Welcome to TiddlyWiki created by Jeremy Ruston, Copyright © 2007 UnaMesa Association
[[Below| prefactorization algebras]] we give a more formal definition, but here we provide the basic idea. Let $M$ be a topological space (which, in practise, will be a smooth manifold).
''Definition''
A prefactorization algebra $\mathcal{F}$ on $M$, taking values in cochain complexes, is a rule that assigns a cochain complex $\mathcal{F}(U)$ to each open set $U \subset M$ along with
* a cochain map $\mathcal{F}(U) \rightarrow \mathcal{F}(V)$ for each inclusion $U \subset V$;
* a cochain map $\mathcal{F}(U_1) \otimes \cdots \otimes \mathcal{F}(U_n) \rightarrow \mathcal{F}(V)$ for every finite collection of open sets where each $U_i \subset V$ and the $U_i$ are pairwise disjoint;
* the maps are compatible in the obvious way (e.g. if $U \subset V \subset W$ is a sequence of open sets, the map $\mathcal{F}(U) \rightarrow \mathcal{F}(W)$ agrees with the composition through $\mathcal{F}(V)$).
//Remark:// A prefactorization algebra resembles a precosheaf, except that we tensor the cochain complexes rather than taking their direct sum.
The observables of a field theory (whether classical or quantum) form a prefactorization algebra on the spacetime manifold $M$. In fact, they satisfy a kind of local-to-global principle in the sense that the observables on a large open set are determined by the observables on small open sets. The notion of a [[factorization algebra]] makes this locality condition precise.
In this appendix, we include a discussion of the tools we use from the theory of nuclear spaces. Throughout this paper, we use cochain complexes of [[nuclear spaces]]. As is well known, homological algebra with topological vector spaces is more subtle than with ordinary vector spaces. Even such basic constructions as taking cohomology can be badly behaved when working with topological vector spaces: the cohomology of a complex of Hausdorff topological vector spaces need no longer be Hausdorff.
Our solution to such problems is to view quasi-isomorphism as too weak a notion of equivalence. Instead, we use cochain homotopy equivalence as our homotopical equivalence relation. Some work is required to show that basic arguments from homological algebra work with this more refined notion of equivalence.
M. F. Atiyah and R. Bott, //A Lefschetz Fixed Point Formula for Elliptic Complexes: I//, The Annals of Mathematics, Second Series, Vol. 86, No. 2 (Sep., 1967), pp. 374-407
In [[Atiyah-Bott]], Atiyah and Bott show that for an elliptic complex $(\E, d)$ on a compact closed manifold $M$, with $\E$ the smooth sections of a $\Z$-graded vector bundle, there is a homotopy equivalence $(\E, d) \hookrightarrow (\br{\E}, d)$ into the elliptic complex of distributional sections. The argument follows from on the existence of parametrices for elliptic operators. We need a simple variant of this result on open sets of a manifold.
Our basic set up is that of a free [[BV theory]]. Simple modifications (along the lines of the proof in [[Atiyah-Bott]]) easily extend this lemma to more general elliptic complexes.
''Lemma''
Let $E \to M$ be a $\Z$-graded vector bundle on a smooth manifold $M$. Let $\E$ denote the sheaf of smooth sections and let $\br{\E}$ denote the sheaf of noncompactly supported distributional sections. Let $Q$ be a degree 1 elliptic operator on $\E$ such that
# $Q^2 = 0$, and
# there exists a degree -1 elliptic operator $Q^*$ such that $[Q,Q^*]$ is a generalized Laplacian on $M$.
Then on any open set $U \subset M$, there is a homotopy equivalence
$$
(\E|_U,Q) \hookrightarrow (\br{\E}|_U,Q).
$$
//Proof//
The proof again relies on the existence of a parametrix to explicitly construct the homotopy equivalence. We will use more standard kernel conventions here (via evaluation pairings) and not the BV conventions we use elsewhere (which depends on a symplectic pairing). This leads to small modifications of the definition we use [[elsewhere|parametrices]]. We just sketch the construction of the necessary parametrix, as it is standard.
Pick a heat kernel $\{K_t\}$ for the generalized Laplacian $D = [Q,Q^*]$ (choose any boundary conditions on M). Let $\phi$ be a cut-off function on $M \times M$ such that $\phi$ is 1 in a small neighborhood of the diagonal and has proper support. In particular, we require $\phi$ to have proper support on $U \times U$. Set
$$
\Phi = \phi \int_0^L Q^* K_t\, dt
$$
for finite $L> 0$. Then $\Phi$ defines a parametrix for $Q$ because
$$
[Q, \Phi] = Q \circ \left( \phi \int_0^L Q^* K_t\, dt \right) + \left( \phi \int_0^L Q^* K_t\, dt \right) \circ Q
$$
$$
= \tilde{\phi} \int_0^L Q^* K_t\, dt + \phi \int_0^L D K_t\, dt
$$
$$
= K_0 - \phi K_L - S
$$
$$
= K_0 - T
$$
where $S$ and $T = \phi K_l + S$ are smoothing operators with proper support away from the diagonal. In the second line, we use the fact that $Q$ commutes with the heat kernel. The term $\tilde{\phi}$ arises because $Q$ is a differential operator, and by the Leibniz rule, there will be a contribution from its (possibly subtle) action on $\phi$. The support of $\tilde{\phi}$, however, is away from the diagonal but proper in $U \times U$ since it only depends on $\phi$. Note that $\phi K_0 = K_0$, since it is the delta function along the diagonal and $\phi$ is 1 in a neighborhood of the diagonal. Moreover, we have shown that the commutator has proper support in $U \times U$ since all the terms do.
The existence of this parametrix with proper support in $U \times U$ gives us a chain homotopy equivalence on the distributional complex $(\br{\E}, Q)$ between the identity and $T$. As $T$ is smoothing, however, the image of $T$ is contained in the image of the inclusion $i: \E \hookrightarrow \br{\E}$. Hence $T$ defines an inverse to $i$, up to homotopy.
$\square$
We define what a BV theory is, as spelled out in \cite{webbook}. All the theories we consider in this wiki are BV theories.
!! A free BV theory
''Definition''
// A // free BV pretheory // on a manifold $M$ consists of the following data:
* a $\Z$-graded super vector bundle $\pi: E \rightarrow M$ that is of finite rank;
* an antisymmetric map of vector bundles $\langle - , - \rangle_{loc} : E \otimes E \rightarrow \op{Dens}(M)$ of degree $-1$ that is fiberwise nondegenerate. It induces a symplectic pairing on compactly supported smooth sections $\E_c$ of $E$:
\[
\langle \phi, \psi \rangle = \int_{x \in M} \langle \phi(x), \psi(x) \rangle_{loc};
\]
* a square-zero differential operator $Q: \E \rightarrow \E$ of cohomological degree 1 that is skew self adjoint for the symplectic pairing.
//
In our constructions, we require the existence of a //gauge-fixing operator// $Q^{GF}: \E \rightarrow \E$ with the following properties:
* it is a square-zero differential operator of cohomological degree $-1$ ;
* it is self adjoint for the symplectic pairing;
* $D=[Q,Q^{GF}]$ is a generalized Laplacian on $M$.
Compare to the [[definition of classical field theory]]. A //free BV theory// consists of a free BV pretheory and a gauge-fixing operator.
The role of the gauge-fixing operator is to construct a heat kernel-type propagator $P$ for the quadratic action functional $\langle \phi, Q \phi \rangle$.
!! A BV theory
Given a free BV theory $(E, \langle -, - \rangle, Q, Q^{GF})$, we can define an interacting theory. The symplectic pairing for the free theory leads to a renormalized version of the usual BV bracket $\{-,-\}_L$ and BV Laplacian $\Delta_L$, where $L$ denotes a length scale. Let $K_L$ denote the kernel $e^{-LD}$. The renormalized BV Laplacian is just $-\partial_{K_L}$. It acts on a functional $I \in \Oo(\E)$ by contraction. We define the BV bracket as the failure of the BV Laplacian to be a derivation:
\[
\{I,J\}_L = \Delta_L(IJ) - (\Delta_L I) J - (-1)^{|I|} I \Delta_L J.
\]
Now we can give a renormalized version of the usual notion of a BV theory. We will give the definition when $M$ is compact; for the sheaf-theoretic version, one needs to make the usual modifications (e.g., use parametrices and so on).
''Definition''
// A // BV theory // consists of a family of interaction terms $I[L] \in \Oo(\E)[[\hbar]]$ parametrized by length scale $L \in (0,\infty)$ satisfying:
* the RG equation $I[L] = W(P(\epsilon,L), I[\epsilon])$ for all $0 < \epsilon, L < \infty$;
* there is a small $L$ asymptotic expansion of $I[L]$ in terms of local functionals;
* the quantum master equation $QI[L] + \frac{1}{2} \{I[L], I[L]\}_L + \hbar \Delta_L I[L] = 0$ for all $L$.
//
Note that the first two conditions are the exact analogues of those for a [[perturbative scalar field theory | quantum field theory]]. The third condition, called the QME, has a natural deformation-theoretic interpretation via our [[main theorem | statement of the deformation quantization theorem]]. It says that our factorization algebra of observables is a BD algebra, turning the $P_0$ factorization algebra of classical observables into a mere factorization algebra, the quantum observables.
We have [[already | P_0 operad]] given a definition of quantization of a $P_0$ algebra. We defined a quantization of a $P_0$ algebra $A$ to be simply an $E_0$ algebra $\til{A}$ over $\R[[\hbar]]$ which reduces to $A$ modulo $\hbar$, and such that a certain bracket constructed from $\til{A}$ coincides (up to homotopy) with the given bracket on $A$.
This definition suffers from some defects: the main one is that there is no reason that the bracket on $A$ induced by $\til{A}$ must be a Poisson bracket.
In this page we will introduce a stronger definition of quantization, where $\til{A}$ is given the structure of an algebra over a certain operad over $\R[[\hbar]]$. This definition is homotopically more natural than the one introduced earlier.
!! The ~Beilinson-Drinfeld operad
Beilinson and Drinfeld constructed an operad over the formal disc which generically is equivalent to the $E_0$ operad, but which at $0$ is equivalent to the $P_0$ operad. We call this operad the ~Beilinon-Drinfeld operad.
The operad $P_0$ is generated by a commutative associative product $-\star-$, of degree $0$; and a Poisson bracket $\{-,-\}$ of degree $+1$.
''Definition''
//The ~Beilinson-Drinfeld (or $BD$) operad is the differential graded operad over the ring $\mathbb{R} [[\hbar]] $ which, as a graded operad, is simply//
$$
BD = P_0 \otimes \mathbb{R}[[\hbar]];
$$
//but with differential defined by//
$$
\mathrm{d} ( -\star - ) = \hbar \{-,-\}.
$$
If $M$ is a flat differential graded $\mathbb{R}[[\hbar]]$ module, then giving $M$ the structure of a $BD$ algebra amounts to giving $M$ a commutative associative product, of degree $0$, and a Poisson bracket of degree $1$, such that the differential on $M$ is a derivation of the Poisson bracket, and the following identity is satisfied:
$$
\mathrm{d} (m \star n) = (\mathrm{d} m ) \star n + (-1)^{\lvert m \rvert} m \star (\mathrm{d} n ) + \hbar \{m,n\}
$$
''Lemma''
//There is an isomorphism of operads,//
$$
BD \otimes_{\mathbb{R}[[\hbar]]} \mathbb{R} \cong P_0,
$$
//and a quasi-isomorphism of operads over $\mathbb{R}((\hbar))$,//
$$
BD \otimes_{\mathbb R[[\hbar]]} \mathbb R((\hbar)) \simeq E_0 \otimes \mathbb{R} ((\hbar)).
$$
Thus, the operad $BD$ interpolates between the $P_0$ operad and the $E_0$ operad.
$BD$ is an operad in the category of differential graded $\mathbb{R}[[\hbar]]$ modules. Thus, we can talk about $BD$ algebras in this category, or in any symmetric monoidal category enriched over the category of differential graded $\mathbb{R}[[\hbar]]$ modules.
The $BD$ algebra is, in addition, a Hopf operad, with coproduct defined in the same way as in the $P_0$ operad. Thus, one can talk about $BD$ factorization algebras.
!! BD quantization of $P_0$ algebras
''Definition''
//Let $A$ be a $P_0$ algebra (in the category of cochain complexes). A BD quantization of $A$ is a flat $\mathbb{R}[[\hbar]]$ module $A^q$, flat over $\mathbb{R}[[\hbar]]$, which is equipped with the structure of a $BD$ algebra, and with an isomorphism of $P_0$ algebras//
$$
A^q \otimes_{\mathbb{R}[[\hbar]]} \mathbb{R} \cong A.
$$
//Similarly, an order $k$ BD quantization of $A$ is a differential graded $\mathbb{R}[[\hbar]]/ \hbar^{k+1}$ module $A^q$, flat over $\mathbb{R}[[\hbar]]/ \hbar^{k+1}$, which is equipped with the structure of an algebra over
the operad//
$$
BD \otimes_{\mathbb{R} [[\hbar]] } \mathbb{R} [[\hbar]] / \hbar^{k+1},
$$
//and with an isomorphism of $P_0$ algebras//
$$
A^q \otimes_{\mathbb{R}[[\hbar]] / \hbar^{k+1}} \mathbb{R} \cong A.
$$
This definition applies without any change in the world of factorization algebras.
''Definition''. //Let $\mc{F}$ be a $P_0$ factorization algebra on $M$. Then a BD quantization of $\mc{F}$ is a BD factorization algebra $\til{\mc{F}}$ equipped with a quasi-isomorphism //
$$
\til{\mc{F}} \otimes_{\R[[\hbar]]} \R \simeq \mc{F}
$$
//of $P_0$ factorization algebras on $M$. //
Berline, Nicole and Getzler, Ezra and Vergne, Michèle, //Heat kernels and Dirac operators//, ~Springer-Verlag 1992
[[Qui69]]
[[Lur10]]
[[Lur09]]
[[Hin01]]
[[Cos11]]
[[Toe06]]
Let $M$ be a manifold and $E$ a graded vector bundle on $M$. Let $U \subset M$ be an open subset. In this page we will introduce some notation for various classes of functionals on sections $\E(U)$ of $E$ on $U$.
!!! Formal power series
The simplest class of functions are the formal power series on $\E(U)$ or on $\E_c(U)$, defined by
$$
\Oo(\E(U)) = \prod_{n \ge 0} \Hom(\E(U)^{\otimes n}, \R)_{S_n}.
$$
Alternatively, we can simply write
$$
\Oo(\E(U)) = \what{\Sym} \E(U)^\vee
$$
where $\E(U)^\vee$ is the strong dual of $\E(U)$, and the completed symmetric algebra $\what{\Sym}$ is taken in the symmetric monoidal category of nuclear spaces.
Note that we can identify
$$
\E(U)^\vee = \br{\E}^!_c (U)
$$
with compactly supported distributional sections of the bundle $E^! = E \otimes \op{Dens}_M$ on $U$.
In a similar way, we define
$$
\Oo(\E_c(U)) = \what{\Sym} \E_c(U)^\vee = \what{\Sym} \br{\E}^!(U).
$$
!!! Functions with proper support
''Definition''
A function $\Phi \in \Oo(\E_c(U))$ has //proper support// if
$$
\d \Phi \in \Oo(\E(U)) \otimes \E_c(U)^\vee \subset \Oo(\E_c(U)) \otimes \E_c(U)^\vee.
$$
The reason for the terminology is as follows. Let $\Phi \in \Oo(\E_c(U))$ and let
$$\Phi_n \in \Hom(\E_c(U)^{\otimes n}, \R)$$
be the $n$'th term in the Taylor expansion of $\Phi$.
Then, $\Phi$ has proper support if and only if, for all $n$, the composition with a projection map
$$
\op{Supp} (\Phi_n ) \subset U^n \to U^{n-1}
$$
is proper.
We will let
$$\Oo^P(\E_c(U)) \subset \Oo(\E_c(U))$$
be the subspace of functions with proper support. Note that functions with proper support are //not// a subalgebra.
Because $\Oo^P(\E_c(U))$ fits into a fibre square
$$
\begin{array} { c c c }
\Oo^P(\E_c(U)) & \to & \Oo(\E(U)) \otimes \E_c(U)^\vee \\
\downarrow & & \downarrow \\
\Oo (\E_c(U)) & \to & \Oo(\E_c(U)) \otimes \E_c(U)^\vee
\end{array}
$$
it has a natural topology, making it into a nuclear space (since it is a closed subspace of the product space $\Oo(\E_c(U)) \oplus \left( \Oo(\E(U)) \otimes \E_c(U)^\vee\right)$).
!!! Functions with smooth first derivative
''Definition''
A function $\Phi \in \Oo(\E_c(U))$ has //smooth first derivative// if $\d \Phi$, which is //a priori// an element of $\Oo(\E_c(U)) \otimes \left(\E_c(U)\right)^{\vee}$, is an element of the subspace
$$
\Oo(\E_c(U)) \otimes \E^!(U).
$$
Note that the space of functions with smooth first derivative is a subalgebra of $\Oo(\E_c(U))$. We will denote this subalgebra by $\Oo^{sm}(\E_c(U))$. Again, the space of functions with smooth first derivative is a nuclear space, because it is defined as the fibre product of a diagram of nuclear spaces.
!!! Functions with smooth first derivative and proper support
We are particularly interested in those functions which have both smooth first derivative and proper support. We will refer to this subspace as $\Oo^{P,sm}(\E_c(U))$. The topology on $\Oo^{P,sm}(\E_c(U))$ is, again, given by viewing as defined by the fibre diagram
$$
\begin{array} { c c c }
\Oo^{P,sm}(\E_c(U)) & \to & \Oo(\E(U)) \otimes \E^!(U) \\
\downarrow & & \downarrow \\
\Oo (\E_c(U)) & \to & \Oo(\E_c(U)) \otimes \br{\E}^!(U)
\end{array}
$$
We have inclusions
$$
\Oo^{sm}(\E(U)) \subset \Oo^{P,sm}(\E_c(U)) \subset \Oo^{sm}(\E_c(U)),
$$
where each inclusion has dense image.
!!! Local functionals
Finally we will describe probably the most important class of functionals, the local functionals
$$
\Ool(\E_c(U)) \subset \Oo(\E_c(U)).
$$
Let $M$ be a manifold, and let $E$ be a graded vector bundle on $M$.
If $U \subset M$ is an open subsets, we will use the following notation for various kinds of sections of $E$ on $U$.
$$
\begin{split}
\E &= \Gamma(M,E) \\
\E(U) &= \Gamma(U,E) \\
\E_c(U) &= \Gamma_c(U,E) \\
\br{\E}(U) &= \text{ distributional sections of } E \text{ on }U \\
\br{\E}_c(U) &= \text{ compactly supported distributional sections of } E \text{ on }U \\
\E^!(U) &= \Gamma(U, E \otimes \op{Dens}_M ) .
\end{split}
$$
In the last definition, $\op{Dens}_M$ refers to the bundle of densities on $M$, and the tensor product is a tensor product of vector bundles. We can define the spaces $\br{\E}^!(U)$, etc. in similar ways. Note that there are various duality relations among these spaces:
$$
\begin{split}
\E(U)^\vee &= \br{\E}^!_c(U) \\
\E_c(U)^\vee &= \br{\E}^!(U) \\
\left(\br{\E}(U)\right)^\vee &= \E^!_c(U) \\
\left( \br{\E}_c(U) \right)^\vee &= \E^!(U).
\end{split}
$$
In all of these expression, $\vee$ refers to the strong dual, and all equalities are isomorphisms of topological vector spaces.
The main aim of this wiki is to present a deformation-quantization approach to quantum field theory. In this page we will outline how a classical field theory gives rise to the classical algebraic structure we consider.
We use the Lagrangian formulation throughout. Thus, classical field theory means the study of the critical locus of an action functional. In fact, we use the language of derived geometry, in which it becomes clear that functions on a [[derived critical locus]] should form a [[P_0 algebra| P_0 operad]], that is, a commutative algebra with a Poisson bracket of cohomological degree $1$. (For an overview of these ideas, see the final section of this page.)
Applying these ideas to infinite-dimensional spaces, such as the space of smooth functions on a manifold, one runs into analytic problems. In particular, the Poisson bracket on classical observables is not always well-defined. There is a large subalgebra of observables, however, on which the Poisson bracket is well-behaved. (This issue leads us to work with [[lax algebras| Lax algebras over an operad]].) Our main theorem about classical field theory is as follows.
''Theorem''
//For a [[classical field theory| definition of classical field theory]] on a manifold $M$, the classical observables $\Obs^{cl}$ form a lax $P_0$ factorization algebra with values in the symmetric monoidal category of cochain complexes of nuclear spaces. //
!! Guide to the section on classical field theory
# Parts (a) - (d) describe our viewpoint on classical field theory as the study of derived critical loci.
# Parts (e) - (i) grapple with the analytic issues, construct the factorization algebra of classical observables, and [[prove|The graded Poisson structure on classical observables]] the the main theorem.
# Parts (j) and (k) discuss further aspects of classical field theory, such as Noether's theorem.
!! A gloss of the main ideas
In the rest of this page, we will outline why one would expect that classical observables should form a $P_0$ algebra. More details are available [[here | Introduction to classical field theory]].
The idea of the construction is very simple: if $U \subset M$ is an open subset, we will let $\mc{EL}(U)$ be the derived space of solutions to the Euler-Lagrange equation on $U$. Since we are dealing with perturbative field theory, we are interested in those solutions to the equations of motion which are infinitely close to a given solution.
The differential graded algebra $\Obs^{cl}(U)$ is defined to be the space of functions on $\mc{EL}(U)$. (Since $\mc{EL}(U)$ is an infinite dimensional space, it takes some work to define $\Obs^{cl}(U)$. Details will be presented [[later | Derived Euler-Lagrange equations]]).
On a compact manifold $M$, the solutions to the Euler-Lagrange equations are the cricital point of the action functional. If we work on an open subset $U \subset M$, this is no longer strictly true, because the integral of the action functional over $U$ is not defined. However, fields on $U$ have a natural foliation, where tangent vectors lying in the leaves of the foliation correspond to variations $\phi \to \phi + \delta \phi$, where $\delta \phi$ has compact support. In this case, the Euler-Lagrange equations are the critical points of a closed one-form $\d S$ defined along the leaves of this foliation.
Any derived scheme which arises as the [[derived critical locus]] of a function acquires an extra structure: it's ring of functions is equipped with the structure of a $P_0$ algebra. The same holds for a derived scheme arising as the derived critical locus of a closed one-form define along some foliation. Thus, we would expect that $\Obs^{cl}(U)$ is equipped with a natural structure of $P_0$ algebra; and that, more generally, the commutative factorization algebrra $\Obs^{cl}$ should be equipped with the structure of $P_0$ factorization algebra.
The infinite dimensional nature of the spaces involved means that it takes some work to define the $P_0$ structure on $\Obs^{cl}(U)$. What we will [[show | the graded poisson structure on classical observables]] is that $\Obs^{cl}(U)$ forms a [[lax | lax algebras over an operad]] $P_0$ algebra, and that $\Obs^{cl}$ forms a factorization algebras valued in lax $P_0$ algebras.
Let $L$ be a local $L_\infty$ algebra on $M$. If $U \subset M$ is an open subset, then $\mscr{L}(U)$ denotes the $L_\infty$ algebra of sections of $L$ on $U$. Let $\L_c(U) \subset \L(U)$ denote the sub-$L_\infty$ algebra of compactly supported sections.
In the [[appendix | Classes of functions on the space of sections of a vector bundle]] we defined the algebra of functions on the space of sections on a vector bundle on a manifold. We are interested in the algebra
$$
\Oo ( \L(U)[1] ) = \prod_{n \ge 0} \Hom \left( (\L(U)[1])^{\otimes n} , \R \right)_{S_n}
$$
where the tensor product is the completed projective tensor product, and $\Hom$ denotes the space of continuous linear maps.
The $L_\infty$ algebra structure on $\L(U)$ gives, as usual, a differential on $\Oo(\L(U)[1])$.
''Definition''
//Define the Lie algebra cochain complex $C^\ast(\L(U))$ to be//
$$C^\ast(\L(U)) = \Oo ( \L(U)[1] )$$
//equipped with the usual ~Chevalley-Eilenberg differential. Similarly, define//
$$
C^\ast_{red}(\L(U)) \subset C^\ast(\L(U))
$$
//to be the reduced ~Chevalley-Eilenberg complex, that is, the kernel of the natural augmentation map $C^\ast(\L(U) \to R$.//
Of course, one can define $C^\ast(\L_c(U)$ in the same way.
We will think of $C^\ast(\L(U))$ as the algebra of functions on the formal moduli problem $B \L(U)$ associated to the $L_\infty$ algebra $\L(U)$.
!!! Cochains with coefficients in a module
Let $L$ be a local $L_\infty$ algebra on $M$, and let $E$ be a graded vector bundle on $M$, equipped with a differential which is a differential operator. As usual, we will let $\mscr{L}$ and $\mscr{E}$ denote the global sections of $L$ and $E$, respectively.
''Definition''
//A local action of $L$ on $E$ is an action of $\mscr{L}$ on $\E$ with the property that the structure maps//
$$
\mscr{L}^{\otimes n} \otimes \E \to \E
$$
//are all polydifferential operators.//
Note that $L^! = L^\vee \otimes_{\cinfty_M} \op{Dens}_M$ has the structure of a local module over $\mscr{L}$.
If $E$ is a local module over $L$, then, for each $U \subset M$, we can define the ~Chevalley-Eilenberg cochains
$$
C^\ast(\L(U), \E(U))
$$
of $\L(U)$ with coefficients in $\E(U)$. As above, one needs to take account of the topologies on the vector spaces $\L(U)$ and $\E(U)$ when defining this ~Chevalley-Eilenberg cochain complex. Thus, as a graded vector space,
$$
C^\ast(\L(U), \E(U)) = \prod_{n \ge 0} \Hom ( (\L(U)[1])^{\otimes n}, \E(U))_{S_n}
$$
where the tensor product is the completed projective tensor product, and $\Hom$ denotes the space of continuous linear maps.
As explained in the [[section on formal moduli problems|Formal moduli problems and Lie algebras]], we should think of a local module $E$ over $L$ as providing, on each open subset $U \subset M$, a vector bundle on the formal moduli problem $B \L(U)$ associated to $\L(U)$. Then the ~Chevalley-Eilenberg cochain complex $C^\ast (\L(U), \E(U))$ should be thought of as the space of sections of this vector bundle.
''Definition''
A //commutative// factorization algebra $F$ on a space $M$ is a factorization algebra such that
* $F$ assigns a commutative algebra $F(U)$ to every open $U \subset M$;
* every structure map is a map of commutative algebras (e.g., for $U \subset V$, the map $F(U) \rightarrow F(V)$ is a morphism of commutative algebras).
This is precisely a [[structured factorization algebra | structured factorization algebras]], with $P = Com$, the commutative operad. We call a factorization $Com$-algebra a commutative factorization algebra, for simplicity.
Let $E,F$ be locally convex Hausdorff topological vector spaces (henceforth, locally convex spaces). Then the completed inductive tensor product $E \br{\otimes} F$ has the universal property that a map
$$
E \br{\otimes} F \to G
$$
to any other locally convex space $G$ is the same thing as a bilinear map
$$
E \times F \to G
$$
which is //separately continuous//, meaning that, for each $f \in F$, the map $E \to G$ obtained by fixing $f$ is continuous, and similarly for each $g \in G$.
Recall that the completed projective tensor product $E \otimes F$ is defined by a similar universal property, except that the map $E \times F \to G$ is required to be continuous for the product topology on $E \times F$. Since this is a stronger condition, we see there is a natural map
$$
E \br{\otimes} F \to E \otimes F.
$$
''Theorem''. //If $E$ and $F$ are both ($\mc{F}$) spaces then the natural map//
$$
E \br{\otimes} F \to E \otimes F
$$
//is an isomorphism of topological vector spaces.//
See [[Grothendieck]], appendix 1.
''Proposition''. // The inductive tensor product commutes with colimits in the category of //complete// locally convex topological vector spaces//.
Again, see [[Grothendieck]] appendix 1.
Although factorization algebras arose by thinking about field theory, they appear naturally in geometry and topology as well. Before explaining the examples, we describe some heuristic reasoning that motivates their construction. The crucial idea is seen quite clearly by considering two disjoint open balls in $M$. A factorization algebra $F$ sends the "sum" $B_1 \sqcup B_2$ to the "product" $F(B_1) \otimes F(B_2)$. In contrast, a cosheaf sends a "sum" to a "sum." Hence, we might think of a factorization algebra as "exponentiating" a cosheaf. While this heuristic has its limitations, it does suggest a way to construct factorization algebras.
A word of warning is in order, however. Satisfying the [[ gluing principle|factorization algebra]] is nontrivial, so although it is easy to see that the examples below form //pre//factorization algebras, they do not always form factorization algebras unless extra hypotheses are included. This issue is, of course, similar to the situation with presheaves and sheaves.
!!!! Symmetric products
Let $X$ be a space. Let $\Sym\, X$ denote the disjoint union of all the symmetric powers $S^d X$ of $X$. We include the "$0^{th}$ symmetric power" $S^0 X = pt$ for formal reasons. (One can view $\Sym\, X$ as the "exponential of $X$".) Notice that $\Sym(X \sqcup Y) = \Sym\, X \times \Sym\, Y$.
The functor sending an open set $U$ to $\Sym\, U$ gives a prefactorization algebra in spaces on the manifold $M$. By composing with the singular chains functor, we get a prefactorization algebra in abelian groups on $M$.
More generally, let $C$ be a symmetric monoidal category where the "symmetric algebra" functor (aka "exponentiation") makes sense. That is, let $C$ be a category where, for any object $X \in C$, we have $\Sym\, X := \coprod_{n \geq 0} (X^{\otimes n})_{S_n}$. Given a cosheaf $\mathcal{F}$ on $M$, the exponential $\Sym \, \mathcal{F}$ is a prefactorization algebra.
!!!! Mapping spaces
Recall that (in the appropriate category of spaces) $Maps(U \sqcup V, X) \cong Maps(U,X) \times Maps(V,X)$. This fact suggests that we might fix a target space $X$ and define a factorization algebra by sending an open set $U$ to $Maps(U,X)$. This construction almost works, but it is not clear how to "extend" a map $f: U \rightarrow X$ from $U$ to a larger open set $ V \ni U$. By working with "compactly-supported" maps, we solve this issue.
Fix $(X,p)$ a pointed space. Let $F$ denote the prefactorization algebra on $M$ sending an open set $U$ to the space of compactly-supported maps from $U$ to $(X,p)$. (Here, "$f$ is compactly-supported" means that the closure of $f^{-1} (X - p)$ is compact.) Then $F$ is a prefactorization algebra in the category of pointed spaces. Composing with the singular chains functor gives a prefactorization algebra in abelian groups.
For a particularly nice example, let $M$ be the circle $S^1$ and $X$ a connected space. Then this factorization algebra essentially encodes the data of the based loop space $\Omega X$.
This prefactorization algebra does not usually satisfy the gluing axiom. By imposing some connectivity hypotheses and working in the $(\infty,1)$-category of spaces, however, we obtain a beautiful [[example|concrete examples of factorization algebras]] of a factorization algebra.
There is a famous theorem of E. Noether that says, in essence, that to every infinitesimal local symmetry of a classical field theory corresponds a "conserved current." We will now describe that statement in our formalism. For simplicity, we will assume that our space-time manifold $M$ is oriented. It is straightforward to modify our constructions to deal with the general case.
In the usual framework, a current $J$ is something which associates to a field $\phi$ an $n-1$ form $J(\phi)$ on space-time, defined up to the addition of an exact $n-1$ form. Thus, we can integrate $J(\phi)$ over any oriented codimension $1$ hypersurface in space-time. The association of $J(\phi)$ to $\phi$ must be local in nature: the value of $J(\phi)$ at a point $p \in M$ must only depend on the values of the derivatives of $\phi$ at $p$.
In our context, the space of fields is described by an elliptic $L_\infty$ algebra $\L$ on space-time $M$. The space of $n-1$ forms is described by the Abelian $L_\infty$ algebra $\Omega^{n-1}_M [-1]$, concentrated in degree $1$. The space of $n-1$ forms modulo exact $n-1$ forms is best described by the truncated de Rham complex $\Omega^{\le n-1}_M[n-2]$, shifted so that $\Omega^{n-1}_M$ is in degree $1$. We will view $\Omega^{\le n-1}_M[n-2]$ as an Abelian local $L_\infty$ algebra.
''Definition.''
//A current on the classical field theory described by $\L$ is a map of local $L_\infty$ algebras //
$$
J : \L \to \Omega^{\le n-1}_M[n-2].
$$
A map of local $L_\infty$ algebras is a map of $L_\infty$ algebras $\L(M) \to \Omega^{\le n-1}_M[n-2]$, with the property that the constituent maps
$$
\L^{\otimes n} \to \Omega^{\le n-1}_M[n-2]
$$
are polydifferential operators.
A map of local $L_\infty$ algebras like this induces a map of sheaves of formal moduli problems. We will let $\op{EL}(U)$ be the formal moduli problem associated to the $L_\infty$ algebra $\L(U)$; so that if $(R,m)$ is a dg Artinian algebra, $\op{EL}(U)(R)$ is the simplicial set of ~Maurer-Cartan elements of $\L(U) \otimes m$.
The formal moduli problem associated to the Abelian $L_\infty$ algebra $\Omega^{\le n-1}(U)[n-2]$ sends a dg Artinian algebra $R$ to the ~Dold-Kan simplicial set of the cochain complex $\Omega^{\le n-1}(U) [n-1] \otimes m$. Recall that $\pi_0$ of this ~Dold-Kan simplicial set is $H^0 ( \Omega^{\le n-1} (U)[n-1] \otimes m ).$ Since $R$ is concentrated in degrees $\le 0$, we see that $\pi_0$ of this simplicial set is
$$
\left( \Omega^{n-1}(U) / \d \Omega^{n-2} (U) \right) \otimes H^0(m).
$$
It follows that if $J$ is a current, then for every dg Artinian ring $(R,m)$, and for every open set $U \subset M$, we get a map
$$
\pi_0 J : \pi_0( \op{EL}(U)(m) ) \to \left( \Omega^{n-1}(U) / \d \Omega^{n-2} (U) \right) \otimes H^0(m).
$$
This map takes a homotopy class of solution to the ~Euler-Lagrange equations on $U$, and yields an $n-1$ form modulo an exact $n-1$ form.
!!! Conserved currents
Suppose that $N \subset M$ is a compact oriented submanifold of codimension one. We will see how we can integrate $J$ a current over $N$ to yield a function on the formal moduli problem $\op{EL}(M)$.
Let $\mbb{A}^1$ denote the formal moduli problem which sends a dg Artinian ring $(R,m)$ to the ~Dold-Kan simplicial set for the cochain complex $m$. The reason for this notation is that $\mbb{A}^1$ is represented by the pro-Artinian algebra $\R[[t]]$. Thus, given any formal moduli problem $F$, a map of formal moduli problems $F \to \mbb{A}^1$ should be regarded as a function on $F$.
Given a compact oriented codimension $1$ submanifold $N$ of $M$, a current $J$ as above yields a map of formal moduli problems
$$
\int_N J : \op{EL}(M) \to \mbb{A}^1 .
$$
This sends an $n$-simplex $\phi \in \op{EL}(M) (R)[n]$ to $\int_N J(\phi)$, which is a closed degree $0$ element of $m \otimes \Omega^\ast(\bigtriangleup^n)$.
Note that $\int_N J(\phi)$ only depends on the behaviour of the field $\phi$ on a neighbourhood of $N$ in $M$. Thus, we will let $\op{EL}(N)$ denote the formal moduli problem of germs of solutions to the ~Euler-Lagrange equation near $N$:
$$
\op{EL}(N)(R) = \liminv_{N \subset U \subset M} \op{EL}(U)(R).
$$
The integral $\int_N J$ is a function on the formal moduli problem $\op{EL}(N)$.
In the usual treatment, a conserved current is a current with the property that, if we let a field evolve from one space-like hypersurface to another, the value of the current does not change.
In order to explain this concept more precisely, let us assume that our space-time manifold is of the form $M = N \times \R$, where $N$ is compact. We will let $N_t$ denote the hypersurface $N \times \{t\}$. Then, for each $t \in \R$ we have a map
$$
\int_{N_t} J : \op{EL}(N_t) \to \mbb{A}^1.
$$
The condition for $\int_{N_t} J$ to be a conserved quantity is that, if $\phi_t \in \op{EL}(N_t)(R)$ is a family of fields which evolve according to the equations of motion, then
$$
\frac{\d}{\d t} \int_{N_t} J (\phi_t) = 0.
$$
To say that a family of fields $\phi_t$ evolves according to the equation of motion means precisely that this family arises from a global solution $\phi \in \op{EL}(M)(R)$.
Thus, the statement that a current is conserved can be recast as saying that, for all solutions $\phi \in \op{EL}(M)(R)$ of the equations of motion, $\int_{N_t} J(\phi)$ is independent of $t$. Of course, this will happen when the $n-1$ form $J(\phi)$ is closed.
A conserved current should thus be a map of local $L_\infty$ algebras from $\L$ to the local $L_\infty$ algebra describing //closed// $n-1$ forms, modulo total derivative. The latter is best described by the Abelian $L_\infty$ algebra $\Omega^{\ast}_M [n-2]$ of all forms, shifted so that $\Omega^{n-1}_M$ is in degree $1$.
''Definition.''
//A conserved current is a map of local $L_\infty$ algebras //
$$
\L \to \Omega^{\ast}_M[n-2].
$$
!!! Noether's theorem
Now we are ready to state Noether's theorem, relating conserved currents and symmetries. We have seen that the dg Lie algebra controlling deformations and symmetries of the classical field theory $\L$ is $\Ool(B \L)$, the local functionals on $\L$.
In our framework, Noether's theorem is a statement about the simplicial set of conserved currents. An $n$-simplex in this simplicial set is simply a family of conserved currents over the base ring $\Omega^\ast(\bigtriangleup^n)$.
''Theorem.''
//There is a natural weak homotopy equivalence between the simplicial set of conserved currents and the ~Dold-Kan simplicial set for the cochain complex $\Ool(B \L)[-1]$. In particular, homotopy classes of conserved currents are in bijection with the group $H^{-1} ( \Ool(B \L))$, which is the group of homotopy classes of infinitesimal symmetries of the classical field theory $\L$.//
''Proof.''
To give a map $\L \to \Omega^{\ast}_M[n-2]$ of local $L_\infty$ algebras is the same as to give a map $J(L) \to J(\Omega^\ast_M)[n-2]$ of $D_M$ $L_\infty$ algebras.
In general, if $\g$ and $\mathfrak{h}$ are $L_\infty$ algebras, the $L_\infty$ algebra controlling maps from $\g$ to $\mathfrak{h}$ is $C^\ast_{red}(\g) \otimes \mathfrak{h}$, the tensor product of the reduced cochain complex of $\g$ with $\mathfrak{h}$.
It follows that to give a map of $D_M$ $L_\infty$ algebras $J(L) \to J(\Omega^\ast_M) [n-2]$ is the same as to give a flat section of
$$
C^\ast_{red}(J(L)) \otimes_{\cinfty_M} J ( \Omega^\ast_M ) [n-1].
$$
The sheaf of flat sections of this can be identified with
$$
\Omega^\ast(M, C^\ast_{red}(J(L)) )[n-1],
$$
the de Rham complex of $M$ with coefficients in the $D_M$-module $C^\ast_{red}(J(L))$, with a shift of $n-1$.
More generally, the simplicial set of maps of local $L_\infty$ algebras $\L \to \Omega^\ast_M[n-2]$ is the ~Dold-Kan simplicial set associated to the cochain complex $\Omega^\ast(M, C^\ast_{red}(J(L)) )[n-1]$.
Since $M$ is oriented, and since $C^\ast_{red}(J(L))$ is a flat $D_M$-module, this cochain coincides with
$$
\omega_M \otimes_{D_M} C^\ast_{red}( J(L )) [-1] = \Ool(B \L ) [-1].
$$
Thus, the simplicial set of conserved currents is the ~Dold-Kan simplicial set for $\Ool (B \L)[-1]$, as desired.
$\square$
Note that is not difficult to modify these constructions to relate "higher conserved currents" (which associate to a field a closed $n-k$-form) and higher infinitesimal symmetries of a classical field theory, corresponding to the groups $H^{n-k} ( \Ool(B \L) )$.
!!! A simple example
Consider the free particle moving in the Euclidean vector space $V \cong \R^n$. We will show how to recover the usual conserved quantity of linear momentum using the formalism described above. We encode this physical system as a free theory on the real line $\R$. Let $V$ also denote the trivial vector bundle over $\R^1$ with fiber $V$.
A section $\phi \in \Gamma(\R^1, V)$ satisfies the equation of motion if $\tfrac{\d^2}{\d t^2} \phi = 0$. Thus, the elliptic $L_\infty$ algebra describing this classical field theory is
$$
\L = \Gamma(\R^1, V)[-1] \xto{\tfrac{\d^2}{\d t^2}} \Gamma(\R^1, V[-2] ) .
$$
This $L_\infty$ is Abelian. The pairing between $\L^1 = \Gamma(\R^1, V)$ and $\L^2 = \Gamma(\R^1, V)$ is
$$
\ip{\phi, \psi} = \int_{\R^1} \ip{\phi(t) , \psi(t)}_V \d t,
$$
where $\ip{-,-}_V$ is the Euclidean inner product on $V$.
We now describe the obstruction-deformation complex $\Ool(B \L)$ for this free theory. Let $J$ denote the infinite jet bundle of smooth functions on $\R^1$. Then the jet bundle $J(V)$ for sections of $V$ is isomorphic to $J \otimes V$. Likewise, recall that $J^\vee := \Hom_{C^\infty}(J,C^\infty) = D_\R$, the ring of differential operators on $\R^1$, and so $J(V)^\vee \cong D \otimes V^\vee$. Hence the $D_\R$ $L_\infty$ algebra associated to $L$ is
$$J(L) = J \otimes (V[-1] \oplus V[-2]).$$
This system is invariant under the translation action of the vector space $V$. We will show how to construct the conserved current associated to translation by each $v \in V$. This conserved current plays the role of momentum.
The Lie algebra of symmetries and deformations of our theory is
$$
\Ool(B \L) [-1] = \omega_{\R^1} \otimes_{D_{\R^1}} C^\ast_{red}( J(L) ) [-1],
$$
which we can identify with
$$
\Omega^\ast (\R^1, C^\ast_{red} ( J(L) ) ).
$$
Since we are dealing with a free field theory, the differential on the Lie algebra cochain complex $C^\ast_{red} ( J(L))$ preserves the grading into homogeneous components. Since momentum is a linear function on the vector space $V$, we are interested in symmetries coming from $J(L)^\vee[-2]$.
The de Rham complex of the $D$ module $J(L)^\vee[-2]$ is concentrated in cohomological degrees $-1$, $0$, and $1$:
$$
0 \rightarrow \Omega^0 \otimes D_{\R^1} \otimes V^\vee \rightarrow (\Omega^0 \otimes D_{\R^1} \otimes V_0^\vee) \oplus (\Omega^1 \otimes D_{\R^1} \otimes V_1^\vee) \rightarrow \Omega^1 \otimes D_{\R^1} \otimes V_0^\vee \rightarrow 0,
$$
and the differential is $\nabla + Q$, where $\nabla$ denotes the flat connection on the bundle $D_{\R^1}$, and $Q$ is the differential on $J(L)^\vee[-1]$.
An element of degree $0$ is a sum of terms
$$
f \otimes \partial^k \otimes v,
$$
with $f$ a smooth function, $\partial = \partial/\partial x$, and $v \in V^\vee$, and of terms
$$
f \, \d x \otimes \partial^k \otimes v.
$$
We compute the differential as follows:
$$
f \otimes \partial^k \otimes v + g\, \d x \otimes \partial^l \otimes w \mapsto (\partial f) \d x \otimes \partial^k \otimes v + f \, \d x \otimes \partial^{k+1} \otimes v + g \, \d x \otimes \partial^{l+2} \otimes w,
$$
since we apply the connection on $D_{\R^1}$ to terms in de Rham degree 0 and the differential on $J(L)^\vee$ to terms in de Rham degree 1.
To each $v \in V$ we have a translation symmetry, given by the closed element of degree $0$
$$X_v = 1 \otimes \partial \otimes v^\vee - \d x \otimes 1 \otimes v^\vee,$$
where $v^\vee \in V^\vee$ is the linear functional given by inner product with $v$.
The associated current $J(X_v)$ simply picks out the $0$-form component of $X_v$. Thus, the current is
$$J(X_v) = 1 \otimes \partial \otimes v.$$
Applied to a field $\phi \in \cinfty(\R^1) \otimes V$, the current $J(X_v)$ yields
$$
\ip{v, \partial \phi}_V \in \cinfty(\R^1).
$$
Evaluated at a point $t \in \R^1$, we find $\ip{v, \partial \phi}_V(t)$, which is the momentum of $\phi$ in the $v$-direction at $t$.
Costello, //A geometric construction of the Witten genus, I//
Costello, Renormalization and effective field theory, 2011
Costello, //A geometric construction of the Witten genus II//, available at http://www.math.northwestern.edu/~costello/elliptic2.pdf
Costello, //Supersymmetric, holomorphic and topological field theories in dimensions 2 and 4//, paper in progress
We have defined a field theory to be a formal elliptic moduli problem equipped with a symplectic form of degree $-1$. The basic way symplectic manifolds arise in geometry is, of course, as cotangent bundles. We can apply this in our setting: given any elliptic moduli problem, we will construct a new elliptic moduli problem -- its shifted cotangent bundle -- which has a symplectic form of degree $-1$. We will call field theories which arise by this construction //cotangent field theories//. It turns out that a surprising number of field theories of interested in mathematics and physics (including, for example, both the $A$- and the $B$-models of mirror symmetry, as well as their half-twisted versions) arise as cotangent theories.
!! The cotangent bundle to an elliptic moduli problem
Let $L$ be an elliptic $L_\infty$ algebra on a manifold $X$; and let $\mc{M}_{L}$ be the associated elliptic moduli problem.
Let $L^!$ be the bundle $L^\vee \otimes \op{Dens}(X)$. Note that there is a natural pairing between compactly supported sections of $L$ and compactly supported sections of $L^!$.
Recall that we use the notation $\mscr{L}$ to denote the space of sections of $L$; we will let $\mscr{L}^!$ denote the space of sections of $L^!$.
''Definition''
//Let us define $T^\ast [k] \mc{M}_L$ to be the elliptic moduli problem associated to the elliptic $L_\infty$ algebra $L \oplus L^! [k-2]$. //
//This elliptic $L_\infty$ algebra has a pairing of cohomological degree $k-2$.//
The $L_\infty$ structure on the space $\mscr{L} \oplus \mscr{L}^! [k-2]$ of sections of the direct sum bundle $L \oplus L^![k-2]$ arises from the natural $\mscr{L}$-module structure on $\mscr{L}^!$.
''Definition''
//Let $\mc M$ be an elliptic moduli problem. Then, the //cotangent field theory// associated to $\mc M$ is the $-1$-symplectic elliptic moduli problem $T^\ast[-1] \mc M$. //
!! Examples
In this section we will list some basic examples of cotangent theories.
In order to make the discussion more transparent, we will normally not explicitly describe the elliptic $L_\infty$ algebra related to an elliptic moduli problem; instead, we will simply define the elliptic moduli problem in terms of the geometric objects it classifies. In all examples, it is straightforward using the techniques we have discussed so far to right down the elliptic $L_\infty$ algebra describing the formal neighbourhood of a point in any of the elliptic moduli problems we will consider.
!!! Self-dual ~Yang-Mills theory
Let $X$ be an oriented $4$-manifold equipped with a conformal class of a metric. Let $G$ be a compact Lie group. Let $\mc M(X, G)$ denote the elliptic moduli problem parametrizing principal $G$-bundles on $X$ with a connection whose curvature is self-dual.
Then, we can consider the cotangent theory $T^\ast[-1] \mc M(X,G)$. This theory is known in the physics literature as \emph{self-dual ~Yang-Mills theory}.
Let us describe the $L_\infty$ algebra of this theory explicitly. Observe that the elliptic $L_\infty$ algebra describing the completion of $\mc M(X,G)$ near a point $(P,\nabla)$ is
$$
\Omega^0(X, \g_P ) \xto{\d} \Omega^1( X, \g_P ) \xto{\d_-} \Omega^2_-(X, \g_P )
$$
where $\g_P$ is the adjoint bundle of Lie algebras associated to the principal $G$-bundle $P$.
Thus, the elliptic $L_\infty$ algebra describing $T^\ast[-1] \mc M$ is given by the diagram
$$
\begin{split}
\Omega^0(X, \g_P ) \xto{\d} & \Omega^1( X, \g_P ) \xto{\d_-} & \Omega^2_-(X, \g_P ) & \\
& \ \ \oplus & \ \oplus \\
& \Omega^2_-(X, \g_P ) \xto{\d} & \Omega^3( X, \g_P ) \xto{\d} & \Omega^4(X, \g_P )
\end{split}
$$
This is a standard presentation of the fields of self-dual ~Yang-Mills theory in the BV formalism.
Ordinary ~Yang-Mills theory arises as a deformation of the self-dual theory. The deformation is given by simply deforming the differential in the dg Lie algebra presented in the diagram above by including a term in the differential which is the identity mapping $\Omega^2_{-}(X,\g_P)$ in degree $1$ to the copy of $\Omega^2_{-}(X,\g_P)$ situated in degree $2$.
!!! Holomorphic ~Chern-Simons theory
Let $E$ be an elliptic curve and let $X$ be a complex manifold. Let $\mc M(E, X)$ denote the elliptic moduli problem parametrizing holomorphic maps from $E \to X$. As before, there is an associated cotangent field theory $T^\ast [-1] \mc M(E,X)$. (In ([[Cos11b]]) it is explained how to describe the formal neighbourhood of any point in this mapping space in terms of an elliptic $L_\infty$ algebra on $E$).
This field theory was called a holomorphic ~Chern-Simons theory in ([[Cos10]],[[Cos11b]]). In the physics literature ([[Wit05]], [[Kap05]]) this theory is known as the $(0,2)$ supersymmetric sigma model.
This theory has an interesting role in both mathematics and physics. For instance, it was shown in ([[Cos10]],[[Cos11b]]) that the partition function of this theory (at least, the part which discards the contributions of non-constant maps to $X$) is the Witten genus of $X$.
!!! Twisted supersymmetric gauge theories
Of course, there are a great many more examples of cotangent theories, as there are very many elliptic moduli problems. In ([[Cos11c]]), it is shown how twisted versions of supersymmetric gauge theories can be written as cotangent theories.
The most basic example is the twisted $\mscr{N}=1$ field theory. If $X$ is a complex surface, and $G$ is a complex Lie group, then the $\mscr{N}=1$ twisted theory is simply the cotangent theory to the elliptic moduli problem of holomorphic principal $G$-bundles on $X$. If we fix one such principal $G$-bundle $P \to X$, then the elliptic $L_\infty$ algebra describing this formal moduli problem near $P$ is
$$
\Omega^{0,\ast}(X, \g_P )
$$
where $\g_P$ is the adjoint bundle of Lie algebras associated to $P$.
The cotangent theory to this elliptic moduli problem is thus described by the elliptic $L_\infty$ algebra
$$
\Omega^{0,\ast} ( X, \g_P \oplus \g_P^\vee \otimes K_X [-1]. ) .
$$
!!! The twisted $\mscr{N}=2$ theory
Twisted versions of gauge theories with more supersymmetry have similar descriptions, as is explained in ([[Cos11c]]). The $\mscr{N}=2$ theory is the cotangent theory to the elliptic moduli problem for holomorphic $G$-bundles $P \to X$ together with a holomorphic section of the adjoint bundle $\g_P$. The elliptic $L_\infty$ algebra describing this moduli problem is
$$
\Omega^{0,\ast} ( X, \g_P + \g_P [-1] ) .
$$
Thus, the elliptic $L_\infty$ algebra for the cotangent theory is
$$
\Omega^{0,\ast} ( X, \g_P + \g_P [-1] \oplus \g_P^\vee \otimes K_X \oplus \g_P^\vee \otimes K_X[-1] ) .
$$
!!! The twisted $\mscr{N}=4$ theory
Finally we will describe the twitsed $\mscr{N}=4$ theory. There are two versions of this twisted theory: one used in the work of ~Vafa-Witten ([[VafWit94]]) on $S$-duality, and another considered more recently by ~Kapustin-Witten ([[KapWit06]]) in their work on geometric Langlands. Here we will describe only the latter.
Let $X$ again be a complex surface, and $G$ a complex Lie group. Then, the twisted $\mscr{N}=4$ theory is the cotangent theory to the elliptic moduli problem describing principal $G$-bundles $P \to X$, together with a holomophic section $\phi \in H^0(X, T^\ast X\otimes \g_P )$, satisfying
$$
[\phi, \phi ] = 0 \in H^0(X, K_X \otimes g_P ) .
$$
Here $T^\ast X$ is the holomoprhic cotangent bundle of $X$.
The elliptic $L_\infty$ algebra describing this is
$$
\Omega^{0,\ast} ( X, \g_P \oplus T^\ast X \otimes \g_P [-1] \oplus K_X \otimes \g_P[-2]) .
$$
Of course, this elliptic $L_\infty$ algebra can be rewritten as
$$
( \Omega^{\ast,\ast}(X, \g_P ) , \dbar )
$$
(so that the differential is just $\dbar$ and does not involve $\partial$). The Lie bracket arises from the commutative algebra structure on the algebra $\Omega^{\ast,\ast}(X)$ of forms on $X$, and the Lie bracket on $\g_P$.
Thus, the elliptic Lie algebra describing the corresponding cotangent theory is
$$
\Omega^{\ast,\ast}(X, \g_P ) \oplus \Omega^{\ast,\ast}(X, \g_P ) [1].
$$
Our definition of local $L_\infty$ algebra is designed to encode derived moduli spaces of solutions to non-linear differential equations. An alternative language for describing differential equations is the theory of D-modules. In this page we will show how our local $L_\infty$ algebras can also be viewed as $L_\infty$ algebras in the symmetric monoidal category of D-modules.
The main motivation for this extra layer of formalism is that local action functionals -- which play a central role in classical field theory -- are elegantly described using the language of D-modules.
Let $C^\infty_M$ denote the sheaf of smooth functions on the manifold $M$, $\op{Dens}_M$ the sheaf of densities, and $D_M$ the sheaf of differential operators. The infinite jet bundle $Jet(E)$ of our vector bundle $E$ is the vector bundle whose fiber at a point $x \in M$ is the space of formal germs at $x$ of sections of $E$. The sheaf of sections of $Jet(E)$, denoted $J(E)$, is equipped with a canonical $D_M$-module structure, i.e., the natural flat connection sometimes known as the Cartan distribution. (For motivation, observe that a field $\phi$ (a section of $E$) gives a section of $Jet(E)$ that encodes all the //local// information about $\phi$. )
The category of $D_M$ modules has a symmetric monoidal structure, given by tensoring over $\cinfty_M$. The following lemma allows us to translate our definition of local $L_\infty$ algebra into the world of D-modules.
''Lemma''
// Let $E_1,\ldots, E_n,F$ be vector bundles on $M$, and let $\mscr{E}_i, \mscr{F}$ denote their spaces of global sections. Then, there is a natural bijection //
$$
\op{PolyDiff}( \mscr{E}_1 \times \dots \times \mscr{E}_n , \mscr{F} ) \iso \op{Hom}_{D_M} (J(E_1)\otimes \dots \otimes J(E_n), J(F) )
$$
//where $\op{PolyDiff}$ refers to the space of polydifferential operators. Further, this bijection is compatible with composition.//
A more formal statement of this lemma is that the multi-category of vector bundles on $M$, with morphisms given by polydifferential operators, is a full subcategory of the symmetric monidal category of $D_M$ modules; with the embedding given by taking jets. The proof of this lemma (which is straightforward) is presented in ([[Cos11]]), Chapter 5.
''Corollary''.
//Let $L$ be a local $L_\infty$ algebra on $M$. Then, $J(L)$ has the structure of $L_\infty$ algebra in the category of $D_M$ modules. //
Indeed, the lemma implies that to give a local $L_\infty$ algebra on $M$ is the same as to give a graded vector bundle $L$ on $M$ together with an $L_\infty$ structure on the $D_M$ module $J(L)$.
We are interested in the Chevalley cochains of $J(L)$, taken in the category of $D_M$ modules. Because $J(L)$ is an inverse limit of the sheaves of finite-order jets, some care needs to be taken when defining this Chevalley cochain complex.
In general, if $E$ is a vector bundle, let $J(E)^\vee$ denote the sheaf $\Hom_{C^\infty_M}(J(E),C^\infty_M)$, where $\Hom_{C^\infty_M}$ denotes continuous linear maps of $C^\infty_M$-modules. This sheaf is naturally a $D_M$-module. We can form the completed symmetric algebra
\[
\Oo_{red}(J(E)) = \prod_{n > 0} \Sym^n \, J(E)^\vee,
\]
which is a $D_M$-algebra.
We should think of an element of $\Oo_{red}(J(E))$ as a Lagrangian on the space $\E$ of sections of $E$. Indeed, a section
$$
F_n \in \Hom_{\cinfty_M} ( J(E)^{\otimes n}, \cinfty_M)
$$
is something which takes $n$ elements $\phi_1,\dots,\phi_n \in \E$ and yields a smooth function $F_n(\phi_1,\dots, \phi_n) \in \cinfty(M)$, with the property that $F_n(\phi_1,\dots, \phi_n)(x)$ only depends on the jet of $\phi_i$ at $x$.
Let $F$ be a section of $\Oo_{red}(E)$, and let us write $F$ as a sum $F = \sum F_n$, where
$$F_n \in \Hom_{\cinfty_M} ( J(E)^{\otimes n}, \cinfty_M) _{S_n}.$$
Then, we can interpret $F$ as something which takes a section $\phi \in \E$ and yields a smooth function
$$
\sum F_n(\phi,\dots,\phi) \in \cinfty(M),
$$
with the property that $F(\phi)(x)$ only depends on the jet of $\phi$ at $x$.
Of course, the functional $F$ is a formal power series in the variable $\phi$. A formal way to say what such a power series is is to use the functor of points: if $R$ is an auxiliary graded Artin ring with maximal ideal $m$, and if $\phi \in \E \otimes m$, then $F( \phi)$ is an element of $\cinfty(M) \otimes m$. This assignment is functorial with respect to maps of graded Artin rings.
!!! Local functionals
We have seen that we can interpret $\Oo_{red}(J(E))$ as the sheaf of Lagrangians on a graded vector bundle $E$ on $M$. Thus, the sheaf
$$
\op{Dens}_M \otimes_{\cinfty_M} \Oo_{red}( J(E))
$$
is the sheaf of Lagrangian densities on $M$. A section $F$ of this sheaf is something which takes a section $\phi \in \E$ of $\E$, and yields a density $F(\phi)$ on $M$, in such a way that $F(\phi)(x)$ only depends on the jet of $\phi$ at $x$. (As before, $F$ is of course a formal power series in the variable $\phi$).
The sheaf of local action functionals is the sheaf of Lagrangians, modulo total derivatives. The formal definition is as follows.
''Definition''
//Let $E$ be a graded vector bundle on $M$, whose space of global sections is $\mscr{E}$. Then, the space of //local action functionals// on $\mscr{E}$ is//
$$
\Oo_{loc}(\E) = \op{Dens}_M \otimes_{D_M} \Oo_{red}( J(E) ).
$$
Here, $\op{Dens}_M$ is the right $D_M$-module of densities on $M$.
Let $\Oo_{red}(\E_c)$ denote the algebra of functionals modulo constants on the space $\E_c$ of compactly supported sections of $E$. Integration over $M$ provides a natural inclusion
$$
\Ool(\E) \to \Oo_{red}(\E_c).
$$
!!! Local Chevalley complex of a local Lie algebra
Let $L$ be a local Lie algebra. Then, we can form, as above, the reduced Chevalley cochain complex $C^\ast_{red}(J(L))$ of $L$. This is the $D_M$-algebra $\Oo_{red}(J(L)[1])$ equipped with a differential encoding the $L_\infty$ structure on $L$.
In general, if $\g$ is an $L_\infty$ algebra, we will think of the Lie algebra cochain complex $C^\ast(\g)$ as being the algebra of functions on $B \g$. Thus, we will let
$$
\Ool(B \mscr{L} ) = \op{Dens}_M \otimes_{D_M} C^\ast_{red}( J( L ) )
$$
denote the space of local action functionals on $J(L)[1]$, equipped with the ~Chevalley-Eilenberg differential. This is the local Chevalley cochain complex. We could also use the notation $C^\ast_{red,loc} ( \L)$ for this complex.
Note that there's a natural inclusion of cochain complexes
$$
\Ool( B \mscr{L} ) \to C^\ast_{red}( \L_c)
$$
where $\L_c$ denotes the $L_\infty$ algebra of compactly supported sections of $L$.
!!! Modules over local $L_\infty$ algebras
Let $L$ be a local $L_\infty$ algebra on $M$, and let $E$ be a local module for $L$. Then, $J(E)$ has an action of the $L_\infty$ algebra $J(L)$, in a way compatible with the $D_M$-module on both $J(E)$ and $J(L)$.
''Definition''
//Suppose that $E$ has a local action of $L$. Then the local cochains $C^\ast_{loc} (\mscr{L}, \E)$ of $\mscr{L}$ with coefficients in $\E$ is defined to be the flat sections of the $D_M$-module of cochains of $J(L)$ with coefficients in $J(E)$. //
More explicitly, the $D_M$-module $C^\ast(J(L), J(E)$ is
$$
\prod_{n \ge 0} \Hom_{\cinfty_M} \left( (J(L)[1]) ^{\otimes n}, J(E) \right)_{S_n}
$$
equipped with the usual ~Chevalley-Eilenberg differential. The sheaf of flat sections of this $D_M$ module is the subsheaf
$$
\prod_{n \ge 0} \Hom_{D_M} \left( (J(L)[1]) ^{\otimes n}, J(E) \right)_{S_n}
$$
where the maps must be $D_M$-linear. In light of the fact that
$$
\Hom_{D_M} \left( J(L)^{\otimes n}, J(E) \right)
= \op{PolyDiff} ( \mscr{L}^{\otimes n}, \E )
$$
we see that $C^\ast_{loc} (\mscr{L}, \E)$ is precisely the subcomplex of the ~Chevalley-Eilenberg cochain complex
$$
C^\ast(\mscr{L}, \E ) = \prod_{n \ge 0} \Hom_{\R} ( (\mscr{L}[1])^{\otimes n}, \E )_{S_n}
$$
consisting of those cochains built up from polydifferential operators.
We defined a classical field theory to be a formal elliptic moduli problem equipped with a symplectic form of degree $-1$. In this page we will rewrite this definition in a more concise (but less conceptual) way. This is included largely for consistency with [[Cos11]], and for ease of reference when we discuss the quantum theory.
''Definition''
//Let $E$ be a graded vector bundle on a manifold $M$. Then, a degree $-1$ symplectic structure on $E$ is an isomorphism of graded vector bundles//
$$
\phi : E \iso E^! [-1]
$$
//which is anti-symmetric, in the sense that $\phi^\ast = -\phi$ where $\phi^\ast$ is the formal adjoint of $\phi$.//
Note that if $L$ is an elliptic $L_\infty$ algebra on $M$ with an invariant pairing of degree $-3$, then the graded vector bundle $L[1]$ on $M$ has a $-1$ symplectic form. Indeed, by definition, $L$ is equipped with a symmetric isomorphism $L \iso L^![-3]$, which becomes an antisymmetric isomorphism $L[1] \iso (L[1])^! [-1]$.
Note also that the tangent space at the basepoint to the formal moduli problem $B \L$ associated to $\L$ is $\L[1]$ (equipped with the differential induced from that on $\L$). Thus, the algebra $C^\ast(\L)$ of cochains of $\L$ is isomorphic, as a graded algebra without the differential, to the algebra $\Oo(\L[1])$ of functionals on $\L[1]$.
Now suppose that $E$ is a graded vector bundle equipped with a $-1$ symplectic form. Let $\Ool(\E)$ denote the space of local functionals on $\E$.
''Proposition.''
#//The symplectic form on $\E$ induces a Poisson bracket on $\Ool(\E)$, of degree $+1$.//
#//To give a local $L_\infty$ algebra structure on $E[1]$, compatible with the given pairing on $E[1]$, is the same as to give an element $S \in \Ool(\E)$ which is of cohomological degree $0$, at least quadratic, and satisfies the classical master equation//
$$
\{S, S\} = 0.
$$
''Proof''.
Let $L = E[-1]$.
Note that $L$ is a local $L_\infty$ algebra, with $0$ differential and bracket. We have seen that the [[exterior derivative|The exterior derivative of a local action functional]] gives a map
$$
\d : \Ool(\E) = \Ool(B \L ) \to C^\ast_{loc} ( L, L^![-1] ) .
$$
Note that the isomorphism
$$
L \iso L^![-3]
$$
gives an isomorphism
$$
C^\ast_{loc} (L, L^![-1] ) \iso C^\ast_{loc} ( L, L [2] ).
$$
Finally, $C^\ast_{loc}(L, L[2])$ is the Lie algebra controlling the deformations of $L$ as a local $L_\infty$ algebra. It thus remains to verify that $\Ool(B \L )\subset C^\ast_{loc}(L, L[2])$ is a sub-Lie algebra; but this is straightforward.
$\square$
Note that the finite-dimensional analog of this statement is simply the fact that, on a formal symplectic manifold, all symplectic derivations (which correspond, after a shift, to deformations of the formal symplectic manifold) are given by Hamiltonian functions, defined up to the addition of an additive constant. The additive constant is not mentioned in our formulation because $\Ool(\E)$ by definition consists of functionals without a constant term.
Thus, we can make a concise definition of a field thoery.
''Definition''
//A pre-classical field theory on a manifold $M$ consists of a graded vector bundle $E$ on $M$, equipped with a symplectic pairing of degree $-1$, and a local functional//
$$
S \in \Ool(\E_c(M))
$$
//of cohomological degree $0$, satisfying the following properties.//
#// $S$ satisfies the classical master equation $\{S,S\} = 0$.//
# //$S$ is at least quadratic (so that $0 \in \E_c(M)$ is a critical point of $S$).//
In this situation, we can write $S$ as a sum (in a unique way)
$$
S(e) = \ip{e, Q e} + I(e)
$$
where $Q : \E \to \E$ is a skew self-adjoint differential operator of cohomological degree $1$ and square zero.
''Definition''
//A pre-classical field is a classical field theory if the complex $(\E,Q)$ is elliptic. //
There is one more property we need of a classical field theories in order to be apply the quantization machinery of ([[Cos11]]).
''Definition''
//A //gauge fixing operator// is a map//
$$
\GF : \E(M) \to \E(M)
$$
//which is a differential operator of cohomological degree $-1$ and square zero, such that//
$$
[Q, \GF] : \E(M) \to \E(M)
$$
//is a generalized Laplacian in the sense of ([[BerGetVer92]])//
The only classical field theories we can try to quantize are those which admit a gauge fixing operator. We will only consider classical field theories which have a gauge fixing operator.
As before, let $M$ be a manifold, $E$ a graded vector bundle on $M$, and $U$ an open subset of $M$. In this page we will define derivations of algebras of functions on $\E(U)$.
''Lemma''
There is an isomorphism (just of vector spaces)
$$
\Oo(\E(U)) \otimes \E(U) \iso \op{Der}(\Oo(\E(U) ) .
$$
where $\op{Der}(\Oo(\E(U) )$ denotes the space of continuous derivations of $\Oo(\E(U))$.
''Proof''
Indeed, any such derivation is determined by where it sends the generators, and is thus determined uniquely by a continuous linear map
$$
(\E(U))^{\vee} \to \Oo(\E(U)).
$$
Now, since $\E(U)^\vee$ is nuclear, the vector space of continuous linear maps is identified with
$$
\E(U) \otimes \Oo(\E(U)).
$$
$\square$
The Lie bracket
$$
\op{Der}(\Oo(\E(U) ) \times \op{Der}(\Oo(\E(U) ) \to \op{Der}(\Oo(\E(U) )
$$
is //not// continuous, but only separately continuous. This means that the map does not extend to a map
$$
\op{Der}(\Oo(\E(U) ) \otimes \op{Der}(\Oo(\E(U) ) \to \op{Der}(\Oo(\E(U) )
$$
from the completed projective tensor product of $\op{Der}(\Oo(\E(U) )$ with itself.
However, the following weaker statement will be useful.
''Lemma.'' The Lie bracket on $\op{Der}( \Oo(\E(U))$ makes $\op{Der}(\Oo(\E(U))$ into a Lie algebra in the symmetric monoidal category of locally topological vector spaces under the [[completed inductive tensor product]].
''Proof.'' This follows immediately from the definition of the completed inductive tensor product, and the fact that the map
$$
\op{Der}(\Oo(\E(U) ) \times \op{Der}(\Oo(\E(U) ) \to \op{Der}(\Oo(\E(U) )
$$
is separately continuous.
$\square$
In this page we will describe the derived version of the moduli space of solutions to the Euler-Lagrange equations, by constructing a Koszul resolution.
Let us recall how one constructs Koszul resolutions for equations in finite dimensions. Let $X$ be a finite dimensional manifold, and let $F$ be a vector bundle on $X$. Suppose that $f \in \Gamma(X,F)$ is a section of $F$, and suppose that we are interested in the derived zero locus of $F$. (In other words, we want to resolve the ring of functions vanishing on the zero locus of $f$.)
We can describe this derived zero locus by a Koszul resolution, as follows. Let us consider the total space of the graded vector bundle $F[-1]$ on $X$. The algebra $\Oo(F[-1])$ of functions on $F[-1]$ is the algebra of sections of the graded algebra $\Sym^\ast (F^\vee [1] )$.
Thus, contraction with the section $f$ of $F$ defines a derivation
$$
\vee f: \Oo(F[-1]) \to \Oo(F[-1])
$$
making $\Oo(F[-1])$ into a differential graded manifold. This differential graded manifold is the derived zero locus of $f$.
We would like to apply this to the Euler-Lagrange equations. The Euler-Lagrange equations are the zero locus of
$$
\d S \in \Oo(\Fields(U)) \otimes \Fields^!(U)
$$
which is a section of the trivial bundle $\Fields^!(U)$ on the formal manifold $\Fields(U)$.
Note that the tensor product here is completed, so that
$$
\Oo(\Fields(U)) \otimes \Fields^!(U) = \prod_{n \ge 0} \Hom ( \Fields(U)^{\otimes n}, \Fields^!(U) )_{S_n}.
$$
Thus, we can define the derived space of solutions to the Euler-Lagrange equation to be the differential graded manifold
$$\Fields(U) \oplus \Fields^!(U)[-1] = \Gamma(U, V \oplus V^![-1]).$$
Functions on this differential graded manifold are
$$
\Oo(\Fields(U) \oplus \Fields^!(U)[-1]) = \prod_{n \ge 0} \Hom ( \left( \Fields(U) \oplus \Fields^!(U)[-1] \right)^{\otimes n}, \R)_{S_n}.
$$
The differential is given by contracting with $\d S$.
''Definition'' The prefactorization algebra of classical observables associated to an action functional $S$ sends an open subset $U$ to
$$
\Obs^{cl} (U) = \Oo ( \Fields(U) \oplus \Fields^!(U)[-1], \d S \vee ).
$$
Note that $\Obs^{cl}(U)$ is a differential graded commutative algebra. Further, if $U \subset V$ are open subsets of $M$, then there is a map of differential graded commutative algebras
$$
\Obs^{cl}(U) \to \Obs^{cl}(V).
$$
Thus, $\Obs^{cl}$ forms a pre-cosheaf of dg commutative algebras.
Any pre-cosheaf of dg commutative algebras can be viewed as a prefactorization algebra. We will see later that $\Obs^{cl}$ is in fact a factorization algebra.
Given a manifold $M$ and a smooth function $f: M \rightarrow \mathbb{R}$, the //critical locus// of $f$ is the subset of points where $df = 0$. Namely, the points where $f$ has extrema, or critical points. For nice functions, the critical locus is a submanifold, although this is not the case for all smooth functions. By adopting the framework of derived geometry, it is possible to study, in a uniform way, the critical locus of any smooth function. More precisely, we work with the //derived// critical locus of a function $f$, which coincides with the usual notion for nice functions but is better behaved for the others.
We explain this notion in the context of finite-dimensional manifolds and then sketch how the derived critical locus appears in classical field theory. Section 4 of the wiki is devoted to making this sketch precise.
!!! The finite-dimensional case
Let $M$ be a compact manifold and $f$ a smooth function on $M$. The critical locus of $f$, $Crit(f)$, is cut out by the equation $\d f = 0$. The ring of functions $\Oo({Crit(f)})$ on this subset is the quotient $\Oo(M)/I_f$, where $I_f$ is the ideal in the ring $\Oo(M) = C^\infty(M)$ generated by the functions $\{X\cdot f \,:\, X \text{ a vector field on } M\}$.
According to the derived philosophy, we should work with an appropriate resolution of this quotient ring that remembers all the relations encoded by the ideal. More precisely, we recognize the zero locus of $\d f$ as the intersection of the graph of $\d f$ in $T^*M$ with the graph of the zero-form (i.e., the natural embedded copy of $M$ in $T^*M$). In the spirit of algebraic geometry, the functions on this intersection is the tensor product $\Oo (graph \, \d f) \otimes_{\Oo(T^* M)} \Oo (M)$. From the derived viewpoint, we ought to work with the derived tensor product $\Oo (graph \, \d f) \otimes^L_{\Oo (T^* M)} \Oo (M)$, which gives the correct answer even when the intersection is not transverse. We view this chain complex as the functions for a differential graded manifold, namely the //derived critical locus// $dCrit(f)$ of $f$.
There is a particularly nice resolution that makes it transparent why $\Oo (dCrit(f))$ is a [[Poisson_0 algebra | P_0 operad]]. Compute the derived tensor product by using the Koszul resolution of $\Oo(M)$ as an $\Oo(T^* M)$ module. We see that $\Oo(dCrit(f))$ is given by the commutative dga
\[
\wedge^{\dim M} T_M \rightarrow \cdots \rightarrow T_M \rightarrow \Oo(M),
\]
where every map is contraction with $\d f$, $T_M$ denotes the tangent sheaf of $M$, and $\wedge^* T_M$ denotes the sheaf of polyvector fields (put in the appropriate cohomological degrees). By construction this is a graded-commutative algebra. It comes with a canonical Poisson bracket of degree 1, known as the Schouten bracket, given by extending the Lie bracket of vector fields in the natural way.
In the language of dg manifolds, this resolution is the shifted cotangent bundle $T^* [-1] M$ equipped with the differential $\d f \vee -$ on $\Oo(T^*[-1]M) = \Sym^* T_M[1]$.
!!! What the derived critical locus sees
The term ``derived" here means that we see the directions in which the naive critical locus deforms. We make this assertion precise as follows. By a //dg artinian algebra//, we mean a graded artinian $\R$-algebra $( \mathscr{A}, \mathfrak{m})$ equipped with a derivation $\partial$, where $\mathfrak{m}$ is the nilpotent maximal ideal and we require $\mathscr{A}/\mathfrak{m} \cong \R$. The set of $\mathscr{A}$-points of $dCrit(f)$ consists of the degree 0 maps of commutative dgas
\[
H^0(\Hom_{cdgas}(\Oo(dCrit(f)), \mathscr{A})),
\]
commuting with the differentials. Such a map $x$ picks out a point in the underlying manifold $M$ by postcomposing with the quotient map $\mathscr{A} \rightarrow \mathscr{A}/\mathfrak{m} \cong \R$. The unreduced part of $x$ encodes how the point can infinitesimally deform over the nilpotent directions of $Spec\, \mathscr{A}$.
!!! ~Euler-Lagrange equations
In the case of classical field theory, our manifold $M$ is in fact an infinite-dimensional vector space, $Fields$, namely the space of smooth sections of some vector bundle. The action functional $S$ plays the role of $f$, and the ideal cutting out the critical locus is generated by the ~Euler-Lagrange equations of $S$. Motivated by the finite-dimensional case, we want to work with the derived critical locus of $S$ instead. There are technical issues in implementing the basic derived strategy when $X$ is an infinite-dimensional manifold, as in our case, but for many kinds of field theories, the classical ~Batalin-Vilkovisky (BV) formalism provides the appropriate method.
!! Gluing
Suppose we have open cover $\{U_i\}$ of $X$ and a factorization algebra $\F_i$ on each $U_i$, with gluing data on double intersections, coherence data on triple intersections, and so on. In this page we will show that we can construct a unique factorization algebra $\F$ on $X$ from this data. In fact, this is an easy consequence of the fact that we can extend factorization algebras from a factorizing basis.
!!! Covers by two open sets
Let us start with the simple case when we cover $X$ by two open sets $U,V$. Let $\F_U$ be a factorization algebra on $U$ and $\F_V$ be a factorization algebra on $V$.
Let $\F_{U \cap V}$ be a factorization algebra on $U \cap V$ equipped with quasi-isomorphisms of factorization algebras on $U \cap V$,
$$
\begin{split}
\F_{U \cap V} &\to i_U^\ast \F_U \\
\F_{U \cap V} &\to i_V^\ast \F_V.
\end{split}
$$
This is one natural way of saying what it means to have "gluing data".
Let us define a factorizing basis $\mf{W}$ of $X$ by saying that a set is in $\mf{W}$ if it lies in $U$ or in $V$.
Let us define an $\mf{W}$-prefactorization algebra $\F$ by saying that
$$
\F(W) = \begin{cases}
\F_U(W) & \text{ if } W \subset U \text{ and } W \not\subset V \\
\F_V(W) & \text{ if } W \subset V \text{ and } W \not\subset U \\
\F_{U\cap V}(W) & \text{ if } W \subset U \cap V
\end{cases}
$$
It is clear that $\F$ is a $\mf{W}$-prefactorization algebra (the structure maps use the quasi-isomorphisms $\F_{U\cap V}(W) \to \F_{U}(W)$ if $W \subset U \cap V$).
''Lemma''. $\F$ is a $\mf{W}$-factorization algebra.
''Proof''. Let $W \in \mf{W}$, and let $\mf{W}'$ be a factorizing cover of $W$ consisting of sets in $\mf{W}$. We need to check that
$$
\check{C} ( \mf{W}', \F) \simeq \F(W).
$$
There are three cases: either $W \subset U \cap V$, or $W \subset U$ and $W \not\subset V$, or $W \subset V$ and $W \not\subset U$. The first case is immediate from the fact that $\F_{U \cap V}$ is a factorization algebra on $U \cap V$.
The second two cases are the same; so let us assume that $W \subset U$ and $W \not \subset V$.
We need to verify that
$$
\check{C} ( \mf{W}', \F) \simeq \F(W).
$$
Now, there are two prefactorization algebras on $U$, $\F$ and $\F_U$. $\F_U$ is given to us, and $\F$ is defined by
$$
\F(W') = \begin{cases}
\F_U(W') & \text{ if } W' \not\subset V \\
\F_{U \cap V} (W') & \text{ if } W' \subset V.
\end{cases}
$$
What we're checking is that $\F$ is a factorization algebra on $U$.
If $W' \in \mf{W}'$, and $W' \subset U \cap V$, then there is a quasi-isomorphism
$$
\F_{U \cap V} (W') \to \F_U (W').
$$
This induces a quasi-isomorphism map of prefactorization algebras on $U$,
$$
\F \to \F_U.
$$
Since $\F_U$ is a factorization algebra, and this map is a quasi-isomorphism, $\F$ is also a factorization algebra.
!!! The general case
Now let us consider a general open cover $\mf{U}$ of $X$. We want to show how to construct, from a factorization algebra on every set in $\mf{U}$ with certain gluing data, a factorization algebra on $X$.
The way we will encode the factorization algebra on the sets in the cover $\mf{U}$, together with gluing data, is as follows. We have, for each finite subset $I \subset \mf{U}$, a factorization algebra $\F_I$ on
$$U_I = \cap_{i \in I} U_i.$$
Further, we have quasi-isomorphisms
$$
\F_{I} \to r_i^\ast \F_{I \setminus \{i\}}
$$
of factorization algebras on $U_I$, for each $i \in I$.
$$
r_i : U_I \to U_{I \setminus \{i\}}
$$
is the natural inclusion.
Finally, we require that, for every $I$ and every $i,j \in I$, the following diagram of commutes:
$$
\begin{array} {c c c }
\F_{I} & \rightarrow & \F_{I \setminus\{i\}} \\
\downarrow & & \downarrow \\
\F_{I \setminus\{j\}} & \rightarrow & \F_{I \setminus\{i,j\} }
\end{array}
$$
We can think of this data as defining a factorization algebra $\F_i$ for each $i \in \mf{U}$, together with quasi-isomorphisms on double intersections, provided by $\F_{\{i,j\}}$; and with coherences provided by the factorization algebras $\F_I$ where $\# I \ge 3$.
In this situation, we can construct a factorization algebra on $X$, as follows. We will let $\mf{W}$ be the factorizing basis of $X$ consisting of open sets subordinate to the cover $\mf{U}$. We will define a $\mf{W}$-factorization algebra $\F$ by saying that
$$
\F(W) = \F_I (W)
$$
where $I$ is the largest subset of $\mf{U}$ such that
$$
W \subset U_I.
$$
It is straightforward to check that $\F$ is a $\mf{W}$-factorization algebra, and so extends to factorization algebra on $X$.
!!! Descent
Let $G$ be a discrete group acting on a space $X$.
''Definition'' A $G$-equivariant factorization algebra on $X$ is a factorization algebra $\F$ on $X$ together with isomorphisms
$$
\rho_g : g^\ast \F \iso \F,
$$
for each $g \in G$, such that
$$
\begin{split}
\rho_{\op{Id}} &= \op{Id} \\
\rho_{g h} &= \rho_h \circ h^\ast (\rho_g ) : h^\ast g^\ast \F \to \F.
\end{split}
$$
''Proposition.'' Let $G$ be a discrete group acting properly discontinuously on $X$, so that $X \to X / G $ is a principal $G$-bundle. Then, there is an equivalence of categories between $G$-equivariant factorization algebras on $X$ and factorization algebras on $X / G$.
''Proof.'' If $\F$ is a factorization algebra on $X / G$, then $f^\ast \F$ is a $G$-equivariant factorization algebra on $X$.
Conversely, let $\F$ be a $G$-equivariant factorization algebra on $\F$. Let $\mf{U}$ be the open cover of $X/G$ consisting of those connected sets where the $G$-bundle $X \to X / G$ admits a section. Note that $\mf{U}$ is a factorizing basis for $X / G$. We will define a $\mf{U}$-factorization algebra $\F^G$ by defining
$$
\F^G (U) = \F( \sigma(U ) )
$$
where $\sigma$ is any section of the $G$-bundle $\pi^{-1}(U) \to U$.
Because $\F$ is $G$-equivariant, $\F (\sigma(U) )$ is independent of the section $\sigma$ chosen. Since $\mf{U}$ is a factorizing basis, $\F^G$ extends canonically to a factorization algebra on $X / G$.
$\square$
Our goal in this section is to explain a simple proposition relating the BV formalism to determinants of complexes. Although the proof is straightforward, it provides an interpretation of the BV formalism as a homological encoding of Wick's lemma: taking a Gaussian integral (the partition function of a free field) recovers the determinant of the quadratic form, and here the BV formalism for a free field returns the determinant of the complex of fields.
!!! Reminder on determinants of complexes
Let $(V,d)$ be a cochain complex whose underlying graded vector space is bounded and finite in every degree.
''Definition''
The determinant of $(V,d)$ is the graded 1-dimensional vector space
$$
\left(\bigotimes_{k \in \Z} \left( \wedge^{\dim V^k} V^k \right)^{(-1)^k} \right) [\chi(V)],
$$
where $L^{-1} := L^\vee$, for $L$ a 1-dimensional vector space.
This determinant is a multiplicative analogue of the Euler characteristic. A crucial property is then the following standard fact.
''Lemma''
There is an isomorphism $\det(V) \cong \det(H^\ast(v))$.
Deligne showed that the Picard groupoid of graded lines is isomorphic to the 1-truncation of the algebraic K-theory spectrum, and that the determinant is isomorphic to the image of the complex in this truncation.
!!! BV formalism and determinants
Let $(V,d)$ be a cochain complex whose underlying graded vector space is bounded and finite in every degree. Let $T^\ast [-1] V$ denote the cochain complex $V \oplus V^\vee[-1]$. This complex has a natural degree -1 symplectic pairing, arising from the canonical evaluation pairing. (Thus it is essentially a free BV theory over a point.) Let $\Oo(T^\ast [-1] V) = \Sym(V^\vee \oplus V[1])$ denote the algebra of functions on $T^\ast[-1] V$. This algebra is equipped with a derivation $d$ by extending the differentials on $V[1]$ and $V^vee$ via the Leibniz rule. However, we equip $\Oo(T^\ast [-1] V)$ with the differential $d + \Delta$, where $\Delta$ is the BV Laplacian. By picking a homogeneous basis $\{x_1,\ldots, x_n\}$ for $V^\vee$, we acquire a dual basis $\{\xi_1,\ldots,\xi_n\}$ for $V[1]$, and with respect to this basis,
$$
\Delta = \sum_j \frac{\partial}{\partial x_i} \frac{\partial}{\partial \xi_i}.
$$
We denote the cochain complex $(\Oo(T^\ast [-1] V), d + \Delta)$ by $\Oo_{BV}(V)$. Note that it is //not// a commutative dg algebra, as $\Delta$ is not a derivation.
In the proposition below, we need to use a peculiar shift of the determinant. For $V$, the Poincare polynomial is
\[
P_V(t) = \sum_k t^k \dim(V^k).
\]
Our shift is
\[
S(V) = -\frac{1}{2}\left( P_V(1) + P_V(-1) \right) + \left[ tP'_V(t) \right]_{t = -1}.
\]
Note that it is an additive invariant of $V$ as it arises from the Poincare polynomial.
''Proposition''
The cohomology of $\Oo_{BV}(V)$ is isomorphic to $\det(V)[S(V)]$.
//Proof://
There is a spectral sequence to compute $H^\ast(\Oo_{BV}(V))$ arising from the filtration $F^k = \Sym^{\leq k} (V^\vee \oplus V[1])$. The first page of this spectral sequence is thus $\Sym(H^\ast(V^\vee) \oplus H^\ast(V)[1])$ with differential given by the induced BV Laplacian $\Delta$. Thus the proposition follows if we prove the proposition when $V$ is just a graded vector space.
For $V$ a graded vector space, the underlying graded algebra of $\Oo_{BV}(V)$ has the form $\C [x_1,\ldots,x_n,\xi_1,\ldots,\xi_n]$, where the $x_k$ are even and the $\xi_k$ are odd. (We obtain this form once we pick a basis for $V$.) A straightforward computation shows that the element $\xi_1 \cdots \xi_n$ generates the cohomology. The shift is another straightforward computation.$\square$
/***
|Name|DiscussionPlugin|
|Source|http://www.TiddlyTools.com/#DiscussionPlugin|
|Documentation|http://www.TiddlyTools.com/#DiscussionPluginInfo|
|Version|1.5.7|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Requires|CommentPlugin|
|Description|display tabbed discussion summary with comment input form|
!!!!!Documentation
>see [[DiscussionPluginInfo]]
!!!!!Configuration
<<<
When installed, [[DiscussionPlugin]] can automatically modify the default shadow [[ViewTemplate]] so that all tiddlers will be rendered with two tabs: "Page", and "Discussion". The "Page" tab displays the regular tiddler content, while the "Discussion" tab displays the summary list of comments as well as an input form to enter new comments. You can enable/disable this action by setting/clearing the following checkbox:
><<option chkDiscussionTemplate>> Automatically modify default shadow [[ViewTemplate]]
Note: //''You must reload your document for changes to this option to take effect.''// In addition, this option is only applied to the shadow [[ViewTemplate]]. If you are using a custom [[ViewTemplate]], you will need to manually alter that template to add the Page and Discussion tab display.
''Please see [[DiscussionPluginInfo]] for additional configuration options and instructions.''
<<<
!!!!!Revisions
<<<
2009.01.04 [1.5.7] in customized ViewTemplate, corrected 'tabs' macro to avoid error when viewing shadow tiddlers
| please see [[DiscussionPluginInfo]] for previous revision details |
2008.04.15 [1.0.0] initial prototype
<<<
!!!!!Code
***/
//{{{
version.extensions.DiscussionPlugin= {major: 1, minor: 5, revision: 7, date: new Date(2009,1,4)};
if (config.options.chkDiscussionTemplate===undefined)
config.options.chkDiscussionTemplate=false;
config.macros.discussion= {
reverse: // display order for summary list
false,
listfmt: // format for summary list items
"#<<slider [[]] [[%tiddler%]] [[%subject%]] [[posted by %who% on %when%]]>>\n",
tags: // tags for comment tiddlers
"excludeLists",
slices: // slice format included in comment tiddlers - used to create summary list display
"/%\n|subject|%subject%|\n|byline|%who%|\n|date|%when%|\n%/",
titlefmt: // format for dynamically generating comment tiddler title
"_%UTC%%random%", // default: append UTC timestamp and random number
commentfmt: // format for individual comment content
"^^posted by %who% on %when%^^\n<<<\n%message%\n<<<\n",
datefmt: // date format for comments
"DDD, MMM DDth, YYYY at hh12:0mm:0ss am",
handler: function(place,macroName,params,wikifier,paramstring,tiddler) {
var here=story.findContainingTiddler(place);
if (here) var tid=here.getAttribute("tiddler"); // containing tiddler title
var listfmt=(params[0]&¶ms[0].length)?params[0]:this.listfmt; // item format
var reverse=(params[1]&¶ms[1].toLowerCase()=="reverse"); if (reverse) params.shift();
var tags=params[1]?params[1]:this.tags; // target tags
if (!tags.readBracketedList().contains("comment")) tags+=" comment"; // must be tagged with "comment"
var commentfmt=(params[2]&¶ms[2].length)?params[2]:this.commentfmt; // output format
var datefmt=(params[3]&¶ms[3].length)?params[3]:this.datefmt; // date format
var tids=store.getTaggedTiddlers("comment","created");
if (reverse||this.reverse) tids=tids.reverse();
var out=""; var count=0;
for (var t=0; t<tids.length; t++) if (tids[t].title!=tid && tids[t].title.substr(0,tid.length)==tid) {
count++;
var title=tids[t].title;
var subject=store.getTiddlerSlice(title,"subject");
var byline=store.getTiddlerSlice(title,"byline");
var when=store.getTiddlerSlice(title,"date");
out+=listfmt;
out=out.replace(/%tiddler%/g,title);
out=out.replace(/%subject%/g,subject);
out=out.replace(/%who%/g,byline);
out=out.replace(/%when%/g,when);
}
out="!!!There "+(count==1?"is ":"are ")+count+" comment"+(count==1?"":"s")+":\n"+out;
var next="%tiddler%"+this.titlefmt;
out+="!!!Add a comment:\n";
out+="<<comment "+next+" [["+tags+"]] [["+this.slices+commentfmt+"]] [["+datefmt+"]]>>";
wikify(out,place);
},
countComments: function(tid,after) {
var tids=store.getTaggedTiddlers("comment","created");
var count=0;
for (var t=0; t<tids.length; t++)
if (tids[t].title!=tid && tids[t].title.substr(0,tid.length)==tid)
if (!after||tid.modified>=after) count++;
return count;
}
};
//}}}
// // automatically add shadow tiddlers and templates for displaying page/discussion tabs
//{{{
// macro for rendering current tiddler content
config.macros.currentTiddler= {
handler: function(place,macroName,params,wikifier,paramstring,tiddler) {
var here=story.findContainingTiddler(place); if (!here) return;
var txt=store.getTiddlerText(here.getAttribute("tiddler"),"");
txt=txt.replace(/\<\<currentTiddler\>\>/g,""); // prevents infinite recursion!
removeChildren(place); wikify(txt,createTiddlyElement(place,"div",null,"viewer"));
}
};
// [[CurrentTiddler]] allows tab to show tiddler content
config.shadowTiddlers.CurrentTiddler="<<currentTiddler>>";
// [[DiscussionTiddler]] allows tab to show discussion panel
config.shadowTiddlers.DiscussionTiddler="<<discussion>>";
// [[NoDiscussionViewTemplate]] is an unmodified copy of the shadow [[ViewTemplate]]
config.shadowTiddlers.NoDiscussionViewTemplate=store.getTiddlerText("ViewTemplate");
// [[DiscussionViewTemplate]] is a copy of the current [[ViewTemplate]] where the
// default viewer content ("view text wikified") is replaced with tabs for Page/Discussion
config.shadowTiddlers.DiscussionViewTemplate=store.getTiddlerText("ViewTemplate").replace(/view text wikified/,
'tabs txtDiscussionTab Page Page CurrentTiddler {{var c=0; if(place) var h=story.findContainingTiddler(place); if(h) c=config.macros.discussion.countComments(h.getAttribute("tiddler")); "Discussion"+(c?" ("+c+")":"")}} Discussion DiscussionTiddler');
// optionally, automatically apply DiscussionViewTemplate to all tiddlers
if (config.options.chkDiscussionTemplate) config.shadowTiddlers.ViewTemplate="[[DiscussionViewTemplate]]";
//}}}
<!--{{{-->
<!--<span macro='getTiddlerPassword'></span>-->
<!--comment the previous line to remove password protection on editing-->
<div class='toolbar' macro='toolbar [[ToolbarCommands::EditToolbar]]'></div>
<div class='title' macro='view title'></div>
<div class='editor' macro='edit title'></div>
<div macro='annotations'></div>
<div class='editor' macro='edit text'></div>
<div class='editor' macro='edit tags'></div><div class='editorFooter'><span macro='message views.editor.tagPrompt'></span><span macro='tagChooser excludeLists'></span></div>
<!--}}}-->
!!! Effective interactions using parametrices
Suppose that $ I[L] $ is a collection of effective interactions defining a theory, as before. We can form an effective interaction $I[\Phi]$ from a parametrix $\Phi$, by the formula
$$
I[\Phi] = W ( P(\Phi)- P(0,L), I[L] ) \in \mathcal{O}(\mathcal{E})[[\hbar]].
$$
This formula makes sense, because $P(\Phi) -P(0,L)$ is a smooth section of $E \boxtimes E$.
Note that the renormalization group equation satisfied by the $I[L]$ implies that $I[\Phi]$ is independent of $L$.
Further, the $I[\Phi]$, as $\Phi$ varies, satisfy their own renormalization group equation,
$$
I[\Phi] = W ( P(\Phi) - P(\Psi), I[\Psi] ).
$$
The $ I[L] $ satisfy a locality axiom, saying that $I[L]$ has a small $L$ asymptotic expansion in terms of local functionals. The functionals $I[\Phi]$ satisfies a similar property.
''Proposition''
//Let us choose (arbitrarily) a metric on $M$, and suppose that that $\Phi$ is a parametrix supported within $\delta$ of the diagonal in $M \times M$. Then there is some constant $c_{i,k}$ (independent of $\Phi$ and $\delta$) such that $I_{i,k}[\Phi]$ is supported within $c_{i,k} \delta$ of the diagonal in $M^k$.//
[[ Proof|proof of locality property for interactions]]
!!! The quantum master equation using parametrices
When we discuss the renormalization group equation based on heat kernels, there is a different quantum master equation for every scale $ L $. In the context we are working in here, there is a different quantum master equation for every parametrix $\Phi$.
Recall that we defined
$$
K_\Phi = K_0 - ([\GF, Q] \otimes 1) \Phi .
$$
Note that $ K_\Phi$ is a smooth section of $ E \boxtimes E $ on $ M \times M $, and that
$$
(Q \otimes 1 + 1 \otimes Q) \left( P(\Phi) - P(\Psi)\right) = K_{\Psi} - K_{\Phi}.
$$
Let
$$
\Delta_{\Phi} : \mathcal{O}( \mathcal{E} ) \rightarrow \mathcal{O} ( \mathcal{E} )
$$
be the operator given by contraction with $ - K_{\Phi} $. Let us define a bracket $ \{-,-\}_{\Phi} $ on $ \mathcal{O}( \mathcal{E}) $ by the formula
$$
\{ \alpha,\beta \}_{\Phi} = \Delta_{\Phi}( \alpha \beta ) - (\Delta_{\Phi} \alpha ) \beta - (-1)^{\abs{\alpha}} \alpha \Delta_{\Phi} \beta.
$$
We say that a functional $ I[\Phi] \in \mathcal{O}(\E)^+ [[\hbar]]$ satisfies the scale $\Phi$ quantum master equation if
$$
Q I[\Phi] + \{I[\Phi], I[\Phi]\}_{\Phi} + \hbar \Delta_{\Phi} I[\Phi] = 0.
$$
If $ I[\Phi] $ satisfies the scale $\Phi$ quantum master equation, then $W( \Psi- \Phi, I[\Phi])$ satisfies the scale $\Psi$ quantum master equation.
Further, if $I[L]$ are a collection of functionals satisfying the heat-kernel version of the renormalization group equation and quantum master equation, and we define functionals $ I[\Phi] $ by
$$
I[\Phi] = W ( P(\Phi) - P(0,L), I[L] )
$$
then each $\Phi$, $ I[\Phi]$ satisfies the scale $\Phi$ quantum master equation.
We are interested in formal moduli problems which describe solutions to differential equations on a manifold $M$. Since we can discuss solutions to a differential equation on any open subset of $M$, such an object will give a (homotopy) sheaf of derived moduli problems on $M$. Let us give a formal definition of such a sheaf.
''Definition.''
//Let $M$ be a manifold. A simplicial presheaf on $M$ is a homotopy sheaf if it satisfies Cech descent. A homotopy sheaf of formal moduli problems on $M$ is a presheaf $F$ of formal moduli problems, with the property that for all Artinian dgas $R$, the simplicial presheaf $F(R)$ is a homotopy sheaf.//
We will often refer to a homotopy sheaf as just a sheaf.
!!! Elliptic moduli problems
We are interested in elliptic derived moduli problems: that is, derived moduli problems described by a system of elliptic partial differential equations on a manifold $M$. We will define a formal pointed elliptic moduli problem on a manifold $M$ to be a sheaf of formal moduli problems represented by a sheaf of $L_\infty$ algebras on $M$ of a certain kind.
''Definition.''
//Let $M$ be a manifold. A local $L_\infty$ algebra on $M$ consists of the following data.//
* //A graded vector bundle $L$ on $M$, whose space of smooth sections will be denoted $\mathcal{L}$. //
* //A differential operator $\d : \mathcal{L} \to \mathcal{L}$, of cohomological degree $1$ and square $0$. //
*// A collection of poly-differential operators//
$$
l_n : \mathcal{L}^{\otimes n} \to \mathcal{L}
$$
//which are alternating, of cohomological degree $2-n$, and which endow $\mathcal{L}$ with the structure of $L_\infty$ algebra.//
''Definition.''
//An elliptic $L_\infty$ algebra is a local $L_\infty$ algebra $\mscr{L}$ as above with the property that $(\mscr{L}, \d)$ is an elliptic complex.//
If $\mathcal{L}$ is a local $L_\infty$ algebra on a manifold $M$, then it yields a presheaf $B \L$ of formal moduli problems on $M$. This presheaf sends a dg Artinian ring $(R,m)$, and an open subset $U \subset M$, to the simplicial set then we can consider the simplicial set
$$
B \L (U)(R) = \op{MC} ( \mathcal{L}(U) \otimes m)
$$
of ~Maurer-Cartan elements of the $L_\infty$ algebra $\mathcal{L}(U) \otimes m$ (where $\mathcal{L}(U)$ refers to the sections of $L$ on $U$). We will think of this as the $R$-points of the formal pointed moduli problem associated to $\mathcal{L}(U)$.
''Definition.''
//A formal pointed elliptic moduli problem (or simply elliptic moduli problem) is a sheaf of formal moduli problems on $M$, which is represented by an elliptic $L_\infty$ algebra.//
The essential data of a classical field theory is the moduli space of solutions to the equations of motion of the field theory. For us, it is essential that we take not the naive moduli space of solutions, but rather the //derived// moduli space of solutions. In the physics literature, the procedure of taking the derived moduli of solutions to the Euler-Lagrange equations is known as the ~Batalin-Vilkovisky formalism.
The derived moduli space of solutions to the equations of motion of a field theory on $X$ is a sheaf on $X$. In this section we will introduce a general language for discussing sheaves of "derived spaces" on $X$ which are cut out by differential equations.
!!! A precise description
[[So far|introduction to classical field theory]], we have a heuristic description of the commutative factorization algebra of observables of a classical field theory.
To make this precise, there are two essential steps.
The first is to work with the [[derived critical locus]] of the action $S$, rather than the usual critical locus. In the physics literature, working with the derived critical locus of the action functional is known as the Batalin-Vilkovisky formalism. Our treatment here follows sections 3 and 6 in chapter 5 of \cite{webbook}. As the procedure is rather abstract, the reader might find it helpful to keep at hand the example of the [[classical free scalar field theory]].
The second step is to pin down the appropriate infinite-dimensional spaces of fields and observables.
Before we proceed, let us first recall the definition of the [[derived critical locus]] of a function $f$ on a finite-dimensional manifold $X$. To take the derived critical locus of $f$, one first passes to the shifted cotangent bundle $T^\ast[-1] X$. The degree $-1$ symplectic form on $T^\ast[-1] X$ induces a Poisson bracket on the algebra $\Oo(T^\ast[-1] X)$ of functions on $T^\ast[-1] X$. Bracketing with the function $f$ (pulled back from $X$) makes $T^\ast[-1] X$ into a differential graded manifold. This differential graded manifold is the derived critical scheme.
We would like to mimic this in the infinite dimensional situation, where the space $X$ has been replaced by the space $\Fields$ of fields, and the function $f$ has been replaced by the action functional $S$.
To do this, however, we need to examine more closely what the ~Euler-Lagrange equations say.
As a first approximation, one can say the following.
:If $M$ is a compact manifold, a field $\phi \in \Fields(M)$ satisfies the ~Euler-Lagrange equation if it is a critical point of the action functional $S$. That is if, for all $\psi\in \Fields(M)$,
$$
S( \phi + \epsilon \psi) = S(\phi) \mod \eps^2.
$$
Note that, for $\psi \in \Fields(M)$, we can write
$$
\frac{\partial S}{\partial \psi} (\phi) = \lim_{\eps \to 0} \frac{1}{\eps} \left( S( \phi + \eps \psi) - S(\phi ) \right).
$$
Since $S$ is really a formal power series on the space of fields, $\frac{\partial S}{\partial \psi} (\phi)$ is also such a formal power series.
We are also interested in solutions to the ~Euler-Lagrange equation in $\Fields(U)$, where $U \subset M$ is an open subset. Note that the action functional $S$ is not defined on $\Fields(U)$, because $S$ involves the integral of a Lagrangian density over the non-compact region $U$.
However, what is defined is the //variation// of the action, but only in compactly supported directions.
Let $\Fields_c(U) \subset \Fields(U)$ denote the subspace of compactly supported fields. We will think of $\Fields_c(U)$ as defining a foliation of the infinite-dimensional formal manifold $\Fields(U)$. Two fields are in the same leaf of this foliation if they differ by a field with compact support.
Note that the action functional $S$ is well-defined on the subspace $\Fields_c(U) \subset \Fields(U)$ of compactly supported fields.
''Lemma''
For all $\psi \in \Fields_c(U)$, the function
$$
\frac{\partial S}{\partial \psi} \in \Oo(\Fields_c(U))
$$
extends uniquely to a function in $\Oo(\Fields(U))$.
Geometrically, this tells us that although the action functional is not defined on the space $\Fields(U)$, we do have a closed one-form $\d S$ on $\Fields(U)$, but only a one-form relative to the foliation given by compactly supported variations of a field. This means that the one-form can only be paired with tangent vectors lying in the leaves of the foliation.
Now, we can state the second approximation to the ~Euler-Lagrange equations.
: A field $\phi \in \Fields(U)$ satisfies the ~Euler-Lagrange equations if, for all $\psi \in \Fields_c(U)$,
$$
\frac{\partial S}{\partial \psi} (\phi) = 0.
$$
Thus, $\phi$ satisfies the ~Euler-Lagrange equations if it is a critical point of the closed leafwise one-form $\d S$.
This version of the ~Euler-Lagrange equations is not quite sufficient for our purposes, either. Let $\br{\Fields}_c(U)$ refer to the distributional completion of the space of compactly supported fields on $U$. That is, $\br{\Fields}_c(U)$ is the space of compactly supported distributional sections of the graded vector bundle $V$ on $U$.
''Lemma'' The map
$$\begin{split}
\Fields_c(U) & \to \Oo(\Fields(U)) \\
\psi & \mapsto \frac{\partial S}{\partial \psi}
\end{split}$$
extends uniquely to a continuous map
$$
\br{\Fields}_c(U) \to \Oo(\Fields(U)).
$$
Now we can state the version of the Euler-Lagrange equations we will use.
''Definition''
A field $\phi\in \Fields(U)$ satisfies the Euler-Lagrange equations if, for all $\psi \in \br{\Fields}_c(U)$,
$$\frac{\partial S}{\partial \psi} (\phi) = 0.$$
It happens that this version of the Euler-Lagrange equations is equivalent to the previous version, for smooth fields $\phi$. One can therefore ask why we need to include distributional variations $\psi$ into our definition.
There are two reasons. First, including distributional variations $\psi$ forces our field $\phi$ to be smooth (without this, one can construct non-smooth solutions to the Euler-Lagrange equations in some cases). The main reason, however, is that ultimately we are interested in the //derived// space of solutions to the Euler-Lagrange equations. Including distributional variations into our definition leads to a better-behaved derived space of solutions.
!!! Flat bundles
Next, let us discuss a more geometric example of an elliptic moduli problem: that describing flat bundles on a manifold $M$. In this case, because flat bundles have automorphisms, it is more difficult to give a direct definition of the formal moduli problem.
Thus, let $G$ be a Lie group, and let $P \to M$ be a principal $G$-bundle equipped with a flat connection. Let $\mf g_P$ be the adjoint bundle (associated to $P$ by the adjoint action of $G$ on its Lie algebra $\g$). Thus, $\g_P$ is a bundle of Lie algebras on $M$, with a flat connection.
Let $R$ be an Artinian dg ring. We want to define the simplicial set $\op{Def}_P(R)$ of $R$-families of flat $G$-bundles on $M$ which deform $P$.
As the underlying topological bundle of $P$ is rigid, we can only deform the flat connection on $P$. A deformation of the connection on $P$ is given by an element
$$
A \in \Omega^1(M, \g_P) \otimes m
$$
of cohomological degree $0$.
We would like to ask that $A$ is flat up to homotopy. The curvature $F(A)$ is
$$
F(A) = \d A + \tfrac{1}{2} [A, A] \in \Omega^2(M, \g_P) \otimes m.
$$
Note that, by the Bianchi identity, $\d F(A) + [A, F(A)] = 0$.
For $A$ to be flat up to homotopy, we should ask that $F(A)$ is exact in the cochain complex $\Omega^2(M, \g_P) \otimes m$ of two-forms on $M$. However, we should also ask that $F(A)$ be made exact in a way compatible with the Bianchi identity.
Thus, as a first approximation, we will define the zero-simplices of the deformation functor by
$$
\op{Def}^{prelim}_P(R)[0] = \{ A \in \Omega^1(M, \g_P) \otimes m, B \in \Omega^2(M, \g_P) \otimes m \mid F(A) = \d_R B , \ \d_{dR} B + [A,B] = 0 \}.
$$
Here, $A$ is of cohomological degree $0$ and $B$ is of cohomological degree $-1$.
Note that if $m$ is of square zero, then the Bianchi constraint on $B$ just says that $\d_{dR} B = 0$. This leads to a problem: the sheaf of closed $2$-forms on $M$ is //not// fine: it has higher cohomology groups. Thus, we cannot hope to construct a deformation functor with values in homotopy sheaves of simplicial sets on $M$ in this way.
Instead, we will ask that $B$ satisfy the Bianchi constraint up a sequence of higher homotopies. Thus, the zero simplices of our simplicial set of deformations are defined by
$$
\op{Def}_P (R)[0] = \{ A \in \Omega^1(M, \g_P) \otimes m, B \in \Omega^{\ge 2}(M, \g_P) \otimes m \mid F(A) + \d_{dR} B + [A,B] + \tfrac{1}{2} [B,B] = 0. \}.
$$
Here, $\d$ refers to the total differential on the tensor product cochain complex $\Omega^{\ge 2}(M, \g_P) \otimes m$. As before, $A$ is of cohomological degree $0$ and $B$ is of cohomological degree $-1$.
If we let $B_i \in \Omega^i (M , \g_P ) \otimes m$, then the first few constraints on the $B_i$ can be written as
$$
\begin{split}
\d_{dR} B_2 + [A, B_2] + \d_R B_3 &= 0 \\
\d_{dR} B_3 + [A, B_3] + \tfrac{1}{2} [B_2, B_2] + \d_R B_4 = 0.
\end{split}
$$
Thus, $B_2$ satisfies the Bianchi constraint up to a homotopy defined by $B_3$, and so on.
The higher simplices of this simplicial set must relate gauge-equivalent flat connections. If the dg ring $R$ is concentrated in degree $0$ (and so has zero differential), then we can define the simplicial set $\op{Def}_P(R)$ to be the homotopy quoient of $\op{Def}_P(R)[0]$ by the nilpotent group associated to the nilpotent Lie algebra $\Omega^0(M, \g_P) \otimes m$, which acts on $\op{Def}_P(R)[0]$ in a standard way.
If $R$ is not concentrated in degree $0$, however, then the higher simplices of $\op{Def}_P(R)$ must also involve elements of $R$ of negative cohomological degree. Indeed, degree $-1$ elements of $R$ should be thought of as homotopies between degree $0$ elements of $R$, and so should contribute $1$-simplices to our simplicial set.
A slick way to define a simplicial set with both desiderata is to set
$$
\op{Def}_P(R)[n] = \{A \in \Omega^\ast(M, \g_P) \otimes m \otimes \Omega^\ast(\bigtriangleup^n ) \mid \d_{dR} A + \d_R A + \tfrac{1}{2} [A,A] = 0 \}.
$$
Suppose that $R$ is concentrated in degree $0$ (so that the differential on $R$ is zero). Then, the higher forms on $M$ don't play any role, and
$$
\op{Def}_P(R)[0] = \{A \in \Omega^!(M, \g_P) \otimes m \mid \d_{dR} A + \tfrac{1}{2} [A,A] = 0 \}.
$$
One can show ([[Get04]]) that the simplicial set $\op{Def}_P(R)$ is weakly homotopy equivalent to the homotopy quotient of $\op{Def}_P(R)[0]$ by the nilpotent group associated to the nilpotent Lie algebra $\Omega^0(M, \g_P) \otimes m$. Indeed, a one-simplex in the simplicial set $\op{Def}_P(R)$ is given by a family of the form $A_0(t) + A_1(t)\d t$, where $A_0(t)$ is a smooth family of elements of $\Omega^1(M, \g_P) \otimes m$ depending on $t \in [0,1]$, and $A_1(t)$ is a smooth family of elements of $\Omega^0(M, \g_P) \otimes m$. The ~Maurer-Cartan equation in this context says that
$$
\begin{split}
\d_{dR} A_0(t) + \tfrac{1}{2} [A_0(t), A_0(t) ] &= 0 \\
\tfrac{\d}{\d t} A_0(t) + [A_1(t), A_0(t) ] &= 0.
\end{split}
$$
The first equation says that $A_0(t)$ defines a family of flat connections. The second equation says that the gauge equivalence class of $A_0(t)$ is independent of $t$. In this way, gauge equivalences are represented by one-simplices in $\op{Def}_P(R)$.
It is immediate that the formal moduli problem $\op{Def}_P(R)$ is represented by the elliptic dg Lie algebra
$$
\mathcal{L} = \Omega^\ast(M,\g).
$$
The differential on $\mathcal{L}$ is the de Rham differential on $M$ coupled to the flat connection on $\g$.
!!! Self-dual bundles
Next, we will discuss the formal moduli problem associated to the self-duality equations on a $4$-manifold. We won't go into as much detail as we did for flat connections; instead, we will simply write down the elliptic $L_\infty$ algebra representing this formal moduli problem.
Let $M$ be an oriented $4$-manifold. Let $G$ be a Lie group, and let $P \to M$ be a principal $G$-bundle, and let $\g_P$ be the adjoint bundle of Lie algebras. Suppose we have a connection $A$ on $P$ with anti-self-dual curvature:
$$F(A)_+ = 0 \in \Omega^2_+(M, \g_P)$$
(here $\Omega^2_+(M)$ denotes the space of self-dual two-forms).
Then, the elliptic Lie algebra controlling deformations of $(P, A)$ is described by the diagram
$$
\Omega^0(M, \g_P ) \xto{\d} \Omega^1(M, \g_P ) \xto{\d_+} \Omega^2_+(M, \g_ P ).
$$
Here $\d_+$ is the composition of the de Rham differential (coupled to the connection on $\g_P$) with the projection onto $\Omega^2_+(M, \g_P)$.
Note that this elliptic Lie algebra is a quotient of that describing the moduli of flat $G$-bundles on $M$.
!!! Holomorphic bundles
In a similar way, if $M$ is a complex manifold and if $P \to M$ is a holomorphic principal $G$-bundle, then the elliptic dg Lie algebra $\Omega^{0,\ast}(M, \g_P)$, with differential $\dbar$, describes the formal moduli space of holomorphic $G$-bundles on $M$.
!!! The free scalar field theory
Let us start with the most basic example of an elliptic moduli problem, that of harmonic functions. Let $M$ be a Riemannian manifold. We want to consider the formal moduli problem describing functions $\phi$ on $M$ which are harmonic, that is, satisfy $\Lap \phi = 0$ where $\Lap$ is the Laplacian. The base point of this formal moduli problem is the zero function.
The elliptic $L_\infty$ algebra describing this formal moduli problem is defined by
$$
\mscr{L} = \cinfty(M) \xto{\Lap} \cinfty(M).
$$
This is situated in degrees $1$ and $2$. The products $l_n$ in this $L_\infty$ algebra are all zero for $n \ge 2$.
In order to justify this definition, let us analyze the ~Maurer-Cartan functor of this $L_\infty$ algebra. Let $R$ be an ordinary (not dg) Artinian ring, and let $m$ be the maximal ideal of $R$. The set of $0$-simplices of the simplicial set $\op{MC}_{\mscr{L}}(R)$ is the set
$$
\{\phi \in \cinfty(M) \otimes m \mid \Lap \phi = 0 .\}
$$
Indeed, because the $L_\infty$ algebra $\mscr{L}$ is Abelian, the set of solutions to the ~Maurer-Cartan equation is simply the set of closed degree $1$ elements of the cochain complex $\mscr{L} \otimes m$. All higher simplices in the simplicial set $\op{MC}_{\mscr{L}}(R)$ are constant. Indeed, if $\phi \in \mscr{L} \otimes m \otimes \Omega^\ast(\bigtriangleup^n)$ is a closed element in degree $1$, then $\phi$ must be in $\cinfty(M) \otimes m \otimes \Omega^0(\bigtriangleup^n)$. The fact that $\phi$ is closed amounts to the statement that $\Lap \phi = 0$ and that $\d_{dR} \phi = 0$, where $\d_{dR}$ is the de Rham differential on $\Omega^\ast(\bigtriangleup^n)$.
Let us now consider the ~Maurer-Cartan simplicial set associated to a differential graded Artinian ring $(R, m)$ with differential $\d_R$. The the set of $0$-simplices of $\op{MC}_{\mscr{L}}(R)$ is the set
$$
\{\phi \in \cinfty(M) \otimes m^0, \ \ \psi \in \cinfty(M) \otimes m^{-1} \mid \Lap \phi = \d_R \psi.\}
$$
(The superscripts on $m$ indicate the cohomological degree). Thus, the zero-simplices of our simplicial set can be identified with the set $R$-valued smooth functions $\phi$ on $M$ which are harmonic up to a homotopy given by $\psi$, and which vanish modulo the maximal ideal $m$.
Next, let us identify the set of $1$-simplices of the ~Maurer-Cartan simplicial set $\op{MC}_{\mscr{L}} ( R)$. This is the set of closed degree $0$ elements of $\mscr{L} \otimes m \otimes \Omega^\ast([0,1])$. Such a closed degree $0$ element has four terms:
$$
\begin{split}
\phi_0(t) & \in \cinfty(M) \otimes m^0 \otimes \Omega^0([0,1]) \\
\phi_1(t) \d t & \in \cinfty(M) \otimes m^{-1} \otimes \Omega^1([0,1]) \\
\psi_0(t) & \in \cinfty(M) \otimes m^{-1} \otimes \Omega^0([0,1]) \\
\psi_1(t) \d t & \in \cinfty(M) \otimes m^{-2} \otimes \Omega^1([0,1])
\end{split}
$$
Being closed amounts to the sequence of equations
$$
\begin{split}
\Lap \phi_0 (t) &= \d_R \psi_0(t) \\
\frac{\d}{\d t} \phi_0(t) = \d_R \phi_1(t) \\
\Lap \phi_1(t) +\frac{\d}{\d t} \psi_0(t) &= \d_R \psi_1(t).
\end{split}
$$
These equations can be interpreted as follows. We think of $\phi_0(t)$ as providing a family of $R$-valued smooth functions on $M$, which are harmonic up to a homotopy specified by $\psi_0(t)$. Further, $\phi_0(t)$ is independent of $t$, up to a homotopy specified by $\phi_1(t)$. Finally, we have a coherence condition among our two homotopies.
The higher simplices of the simplicial set have a similar interpretation.
!!! Interacting scalar field theories
Next, we will consider an elliptic moduli problem which arises as the ~Euler-Lagrange equation for an interacting scalar field theory. The ~Euler-Lagrange equation for the action functional $\tfrac{1}{2}\int \phi \Lap \phi + \tfrac{1}{4!}\phi^4$ (where $\phi \in \cinfty(M)$ is a smooth function) is the equation
$$
\Lap \phi + \tfrac{1}{3!} \phi^3 = 0.
$$
The formal moduli problem of solutions to this equation can be described as the solutions to the ~Maurer-Cartan equation in a certain elliptic $L_\infty$ algebra which (as always) we call $\L$. As a cochain complex, $\L$ is
$$
\L = \cinfty(M)[-1] \xto{\Lap} \cinfty(M)[-2].
$$
Thus, $\cinfty(M)$ is situated in degrees $1$ and $2$, and the differential is the Laplacian.
The $L_\infty$ brackets $l_n$ are all zero except for $l_3$. The cubic bracket $l_3$ is the map
$$
\begin{split}
l_3 : \cinfty(M)^{\otimes 3} &\to \cinfty(M) \\
\phi_1 \otimes \phi_2 \otimes \phi_3 & \mapsto \phi_1 \phi_2 \phi_3.
\end{split}
$$
Here, the copy of $\cinfty(M)$ appearing in the source of $l_3$ is the one situated in degree $1$, whereas that appearing in the target is the one situated in degree $2$.
If $R$ is an ordinary (not dg) Artinian ring, then the ~Maurer-Cartan simplicial set $\op{MC}_{\mscr{L}}(R)$ associated to $R$ has for $0$-simplices the set $\phi \in \cinfty(M) \otimes m$ such that $\Lap \phi + \tfrac{1}{3!} \phi^3 = 0$. The higher simplices of this simplicial set are constant.
If $R$ is a dg Artinian ring, then the simplicial set $\op{MC}_{\mscr{L}}(R)$ has for zero simplices the set of pairs $\phi \in \cinfty(M) \otimes m^0$ and $\psi \in \cinfty(M) \otimes m^{-1}$ such that
$$
\Lap \phi + \tfrac{1}{3!} \phi^3 = \d_R \psi.
$$
We should interpret this as saying that $\phi$ satisfies the ~Euler-Lagrange equations up to a homotopy given by $\psi$.
The higher simplices of this simplicial set have an interpretation similar to that described for the free theory.
Let us now give some basic examples of field theories arising as the derived critical locus of an action functional. We will only discuss scalar field theories in this section.
Let $M$ be a Riemannian manifold. Let $E = \underline{\R}$ be the trivial vector bundle on $M$, and let
$$
S(\phi ) = \tfrac{1}{2} \int_M \phi \Lap \phi
$$
denote the action functional for the free massless field theory on $M$. Here $\op{Lap}$ is the Laplacian on $M$, viewed as a differential operator from $\cinfty(M)$ to $\op{Dens}(M)$.
The derived critical locus of $S$ is described by the elliptic $L_\infty$ algebra
$$
\L = \cinfty(M)[-1] \xto{\Lap} \op{Dens}(M)[-2]
$$
where $\op{Dens}(M)$ is the global sections of the bundle of densities on $M$. Thus, $\cinfty(M)$ is situated in degree $1$, and the space $\op{Dens}(M)$ is situated in degree $2$. The pairing between $\op{Dens}(M)$ and $\cinfty(M)$ gives the invariant pairing on $\L$, which is symmetric of degree $-3$ as desired.
!!! Interacting scalar field theories
Next, let us write down the derived critical locus for a basic interacting scalar field theory, given by the action functional
$$
S(\phi ) = \tfrac{1}{2} \int_M \phi \Lap \phi + \tfrac{1}{4!} \int_M \phi^4 .
$$
The cochain complex underlying our elliptic $L_\infty$ algebra is, as before,
$$
\L = \cinfty(M)[-1] \xto{\Lap} \op{Dens}(M)[-2].
$$
The interacting term $\tfrac{1}{4!} \int_M \phi^4$ gives rise to a higher bracket $l_3$ on $\L$, defined by the map
$$
\begin{split}
\cinfty(M)^{\otimes 3} & \to \op{Dens}(M) \\
\phi_1 \otimes \phi_2 \otimes \phi_3 & \mapsto \phi_1 \phi_2 \phi_3 \d Vol.
\end{split}
$$
Let $(R,m)$ be a nilpotent Artinian ring, concentrated in degree $0$. Then, a section of $\phi \in \cinfty(M) \otimes m$ satisfies the ~Maurer-Cartan equation in this $L_\infty$ algebra if and only if
$$
\Lap \phi + \tfrac{1}{3!} \phi^3 \d Vol = 0.
$$
Note that this is precisely the ~Euler-Lagrange equation for $S$. Thus, the formal moduli problem associated to $\L$ is, as desired, the derived version of the moduli of solutions to the ~Euler-Lagrange equations for $S$.
!! Factorization algebras defined on a factorizing basis
Let $X$ be a topological space, and let $\mf{U}$ be a basis for $X$, which is closed under taking finite intersections. It is well-known that there is an equivalence of categories between sheaves on $X$ and sheaves which are only defined for open sets in the basis $\mf{U}$. In this section we will prove a similar statement for factorization algebras. This will allow us to perform several useful formal constructions with factorization algebras, such as gluing.
''Definition'' A //factorizing basis// for $X$ is a basis $\mf{U}$ of open sets of $X$ which is closed under finite intersections, and which is also a factorizing cover.
Let $\mf{U}$ be a factorizing basis.
''Definition''
A $\mf{U}$-prefactorization algebra $\F$ is like a factorization algebra, except that $\F(U)$ is only defined for $U \in \mf{U}$. A $\mf{U}$-factorization algebra is a $\mf{U}$-prefactorization algebra with the property that, for all $U \in \mf{U}$ and all factorizing covers $\mf{V}$ of $U$ consisting of open sets in $\mf{U}$,
$$
\check{C}( \mf{V}, \F) \simeq \F(U),
$$
where $\check{C}(\mf{V}, \F)$ denotes the Cech complex described [[earlier | factorization algebra]].
In this page we will show that any $\mf{U}$-factorization algebra on $X$ extends to a factorization algebra on $X$. This extension is unique up to quasi-isomorphism.
Let $\F$ be a $\mf{U}$-factorization algebra. Let us define a prefactorization algebra $i_\ast^{\mf{U}} \F$ on $X$ by
$$
i_\ast^{\mf{U}}(\F) (V) = \check{C}(\mf{U}_V, \mc{F}),
$$-
for each $V \subset X$ open. Here $\mf{U}_V$ is the cover of $V$ consisting of those open subsets in the cover $\mf{U}$ which are contained in $V$.
''Lemma.''
With this definition, $i_\ast^{\mf{U}}(\F)$ is a factorization algebra whose restriction to open sets in the cover $\mf{U}$ is quasi-isomorphic to $\F$.
''Proof.'' We need to check that if $\mf{W}$ is a factorizing cover of $V \subset X$, then
$$
i_\ast^{\mf{U}}(\F) (V) \simeq \check{C}(\mf{W}, i_\ast^{\mf{U}}(\F) ).
$$
Before we prove this, we need a lemma. Let $\mf{U}_{\mf{W}}$ be the cover of $V$ consisting of open sets in $\mf{U}$ which are subordinate to $\mf{W}$.
''Sub-lemma''. For any $\mf{U}$-prefactorization algebra $\F$, the natural map
$$
\check{C}(\mf{W}, i_\ast^{\mf{U}}(\F) ) \to \check{C}(\mf{U}_{\mf{W}}, \F)
$$
is a quasi-isomorphism.
[[Proof | Proof of Cech lemma]]
It remains to shows that the natural map
$$
\check{C}(\mf{U}_{\mf{W}}, \F) \to \check{C}(\mf{U}_V, \F)
$$
is a quasi-isomorphism. (Here, as before, $\mf{U}_V$ refers to the cover of $V$ consisting of sets in $\mf{U}$ which lie in $V$).
To see that this map is a quasi-isomorphism, observe that by another application of the sub-lemma there is a quasi-isomorphism
$$
\check{C}( \mf{U}, i_\ast^{\mf{U}_{\mf{W}}}(\F) ) \simeq \check{C} (\mf{U}_{\mf{W}}, \F).
$$
Here $i_\ast^{\mf{U}_{\mf{W}}}$ refers to the prefactorization algebra on $V$ obtained by extending $\F$, as before, but now considered as a $\mf{U}_{\mf{W}}$-factorization algebra.
Now the fact that $\F$ is a $\mf{U}$-factorization algebra implies that, for all $U \in \mf{U}$, the natural map
$$
\check{C} ( \mf{U}_{\mf{W}} \cap \mf{U}_U, \F) \to \F(U)
$$
is a quasi-isomorphism.
It follows that the natural map
$$
\check{C}( \mf{U}, i_\ast^{\mf{U}_{\mf{W}}}(\F) ) \to \check{C}( \mf{U}, \F )
$$
is a quasi-isomorphism, as desired.
$\square$
A factorization algebra is a prefactorization algebra that satisfies the //locality// axiom. This axiom is the analog of the gluing axiom for sheaves; it expresses how the values on big open sets are determined by the values on small open sets. For sheaves, the gluing axiom says that for any open set $U$ and any cover of that open set, we can determine the value of the sheaf on $U$ from the values on the open cover. For factorization algebras, we require our covers to be fine enough that they capture all the "multiplicative structure."
We will describe the locality axiom for factorization algebras taking values in vector spaces or chain complexes, but the generalization to an arbitrary symmetric monoidal category is straightforward.
''Definition''
//Let $U$ be an open set and $\mathfrak{U} := \{ U_i \mid i \in I\}$ a cover of $U$ by open sets. The cover $\mathfrak{U}$ is //factorizing// if for any finite collection of points $\{x_1,\ldots,x_k\}$ in $U$, there is a finite collection of pairwise disjoint opens $\{U_{i_1}, \ldots, U_{i_n}\}$ from the cover such that $\{x_1,\ldots,x_k\} \subset U_{i_1} \cup \cdots \cup U_{i_n}$.//
//Remark:// Every Hausdorff space admits a nontrivial factorizing cover (i.e., a cover not containing the whole space as an element).
//Remark:// For a smooth $n$-manifold $M$, there is a simple way to construct a factorizing basis for $M$. Namely, fix a Riemannian metric on $M$, and consider
\[
\{ B_r(x) \,:\, \forall x \in M, \text{ with } 0 < r < InjRad(x)\},
\]
the collection of open balls, running over each point $x \in M$, whose radii are less than the injectivity radius at $x$. Another construction is simply to take the collection of open sets in $M$ diffeomorphic to the open $n$-ball.
!!! Strict factorization algebras
The value of a factorization algebra on $U$ is determined by its behavior on a factorizing cover, just as the value of a cosheaf on an open set $U$ is determined by its value on any cover of $U$.
In order to motivate our definition of factorization algebra, let us write briefly recall the cosheaf axiom. A precosheaf $\Phi$ on $M$ is a cosheaf if, for every open cover $\{U_i \mid i \in I\}$ of an open set $U \subset M$, the sequence
$$
\oplus_{i,j} \Phi (U_i \cap U_j ) \to \oplus_k \Phi(U_k) \to \Phi(U)
$$
is exact on the right. (Alternatively, one can say the map $\oplus \Phi(U_k) \to \Phi(U)$ coequalizes the pair of maps $\oplus \Phi(U_i \cap U_j) \rightrightarrows \oplus \Phi(U_k)$.)
We will define the notion of factorization algebra in a similar way, except that instead of considering elements $U_i$ of the cover, one considers finite collections of disjoint elements of the cover.
In order to make this precise, we need to introduce some notation. Let $P I$ denote the set of finite subsets $\alpha \subset I$, with the property that if $j,j' \in \alpha$, $U_j \cap U_{j'} = \emptyset$. These are the tuples of open subsets that appear in the structure maps of a prefactorization algebra.
If $\alpha \in PI$, let us define $\F(\alpha)$ by
$$
\F(\alpha) = \otimes_{j \in \alpha} \F( U_j ).
$$
Similarly, if $\alpha_1, \ldots, \alpha_k \in P I$, we will let
$$
\F(\alpha_1, \ldots, \alpha_k ) = \otimes_{j_1 \in \alpha_1,\ldots, j_k \in \alpha_k} \F (U_{j_1} \cap \cdots \cap U_{j_k}).
$$
Note that there are natural maps
$$
p_i : \F( \alpha_1,\ldots, \alpha_k ) \to \F( \alpha_1, \ldots, \what{\alpha_i}, \ldots, \alpha_k )
$$
for each $1 \le i \le k$.
''Definition''
//A prefactorization algebra is a //strict factorization algebra// if, for every open subset $U \subset M$ and every factorizing cover $\{U_i \mid i \in I\}$ of $U$, the sequence//
$$
\oplus_{\alpha_1, \alpha_2 \in P I} \F( \alpha_1, \alpha_2 )\xto{p_1 - p_2} \oplus_{\beta \in P I } \F(\beta) \to \F(U)
$$
//is exact on the right. //
!!! The Cech complex and homotopy factorization algebras
Now suppose we have a prefactorization algebra $\F$, taking values in complexes. We will define what it means for $\F$ to be a //homotopy factorization algebra//. This will happen when $\F(U)$ is quasi-isomorphic to a certain Cech complex constructed from any factorizing cover.
To motivate the definition, let us first recall the definition of a homotopy cosheaf. Let $\Phi$ be a pre-cosheaf on $M$, and let $\mf{U} = \{U_i \mid i \in I\}$ be a cover of some open subset $U$ of $M$. The Cech complex of $\mf{U}$ with coefficients in $\Phi$ is is defined in the usual way, as
$$
\oplus_k \oplus_{j_1,\ldots, j_k \in I} \Phi (U_{j_1} \cap \cdots \cap U_{j_k} ) [ k - 1]
$$
where the differential is defined in the usual way. We say that $\Phi$ is a homotopy cosheaf if the natural map from the Cech complex to $\Phi(U)$ is a quasi-isomorphism, for every open $U \subset M$ and every open cover of $U$.
Now let $\F$ be a prefactorization algebra on $M$, and let $\mf{U} = \{U_i \mid i \in I\}$ be a factorizing cover of an open subset $U \subset M$. The Cech complex of $\mf{U}$ with coefficients in $\F$ is defined by
$$
\check{C} ( \mf{U}, \F) = \oplus_{k \ge 0} \oplus_{\alpha_1,\ldots, \alpha_k \in P I } \F ( \alpha_1, \ldots, \alpha_k) [k-1]
$$
with differential defined as the alternating sum of the restriction maps $\F(\alpha_1, \ldots, \alpha_k) \to \F( \alpha_1, \ldots, \what{\alpha_i}, \ldots, \alpha_k )$, as in the Cech complex for a homotopy cosheaf.
''Definition.''
//A homotopy factorization algebra on $X$ is a prefactorization algebra $\F$ valued in cochain complexes, with the property that for every open set $U \subset X$, factorizing cover $\mf{U}$ of $U$, the natural map//
$$
\check{C}(\mf{U},\F) \to \F(U)
$$
//is a quasi-isomorphism.//
//Remark.// The notion of strict factorization algebra is not appropriate for the world of cochain complexes. Whenever we refer to a factorization algebra in cochain complexes, we will mean a homotopy factorization algebra.
!!! Factorization algebras in quantum field theory
We have [[seen | Prefactorization algebras in quantum field theory]] how prefactorization algebras appear naturally when one thinks about the structure of observables of a quantum field theory. It is natural to ask whether the locality axiom which distinguishes factorization algebras from prefactorization algebras also has a quantum-field theoretic interpretation.
The locality axiom we posit states, roughly speaking, that all observables on an open set $U \subset M$ can be built up as sums of observables supported on arbitrarily small open subsets of $M$. To be concrete, let us consider a factorizing cover $\mf{U}_\eps$ of $M$, consisting of all balls in $M$ of radius $< \eps$. Applied to this factorizing cover, our locality axiom states that any observable $O \in \Obs(U)$ can be written as a sum of observables of the form $O_1 O_2 \cdots O_k$, where $O_i \in \Obs(B_{\delta_i} (x_i))$ and $x_1,\ldots, x_k \in M$.
By taking $\eps$ to be very small, we see that our locality axiom implies that all observables can be written as sums of products of observables which are supported as close as we like to points in $U$.
This is a physically reasonable assumption: most of the observables (or operators) which are considered in quantum field theory textbooks are supported at points, so it might make sense to restrict attention to observables built from these.
However, more global observables are also considered in the physics literature. For example, in a gauge theory, one might consider the observable which measures the monodromy of a connection around some loop in the space-time manifold. How would such observables fit into the factorization algebra picture?
The answer reveals a key limitation of our axioms: //the concept of factorization algebra is only appropriate for perturbative quantum field theories//. Indeed, in a perturbative gauge theory, the gauge field (i.e., the connection) is taken to be an infinitesimally small perturbation $A_0 + \delta A$ of a fixed connection $A_0$, which is a solution to the equations of motion. There is a well-known formula (the time-ordered exponential) expressing the holonomy of $A_0 + \delta A$ as a power series in $\delta A$, where the coefficients of the power series are given as integrals over $L^k$, where $L$ is the loop which we are considering.
This expression shows that the holonomy of $A_0 + \delta A$ can be built up from observables supported at points (which happen to lie on the loop $L$). This, the holonomy observable will form part of our factorization algebra.
However, if we are not working in a perturbative setting, this formula does not apply, and we would not expect (in general) that the prefactorization algebra of observables satisfies the locality axiom.
The goal of this section is to describe a natural class of factorization algebras. The factorization algebras which we construct from classical and quantum field theory will be closely related to the factorization algebras discussed here.
The main result of this page is that, given a nice cosheaf of vector spaces $F$, the functor $\Sym^\ast F: U \mapsto \Sym(F(U))$ is a factorization algebra. It is clear how this functor is a prefactorization algebra; the hard part is verifying when it satisfies the locality axiom. We will pin down properties of a cosheaf that guarantee $\Sym^\ast F$ is a factorization algebra.
We begin by providing the definitions necessary to state the main result of this section. We then state the main result and explain its role for the rest of the wiki. Finally, we prove the lemmas that culminate in the proof of the main result.
!!! Preliminaries
Many of the definitions below are the obvious analogues for cosheaves of definitions in sheaf theory.
Throughout, let $M$ denote a locally compact, paracompact, Hausdorff space. Let $R$ be a commutative nuclear algebra (over $\R$ or $\C$). Let $R-\op{mod}$ the category of projective nuclear $R$-modules. When equipped with the completed projective tensor product over $R$, $R-\op{mod}$ becomes a symmetric monoidal category.
''Definition'' Let $F$ be a precosheaf, valued in nuclear spaces or cochain complexes of nuclear spaces. Let $\Sym^\ast F$ denote the prefactorization algebra that assigns to any open set $U$ the symmetric algebra
$$\Sym^\ast(F(U)) = \oplus_{k \ge 0} \Sym^k F(U).$$
Let $\what{\Sym}^\ast F$ denote the prefactorization algebra which assigns to $F$ the completed symmetric algebra
$$
\what{\Sym}^\ast F(U) = \prod_{k \ge 0} \Sym^k F(U).
$$
In this case of the completed symmetric algebra, we view the prefactorization algebra as taking values in the category of pro-cochain complexes of nuclear spaces, with the natural pro-structure $\liminv \Sym F(U)/\mathfrak{m}^n$, where $\mathfrak{m}$ denotes the ideal generated by $F(U)$.
!!! The locality axiom
Now we can state the main results of this page.
''Proposition''
# If $F$ is a cosheaf of nuclear spaces, then $\Sym^\ast F$ is a factorization algebra, as is the completed symmetric algebra $\what{\Sym}^\ast F$ (where $\what{\Sym}^\ast F$ is viewed as a pro-cochain complex of nuclear spaces).
# If $F$ is a homotopy cosheaf of cochain complexes of nuclear spaces, then both $\Sym^\ast F$ and $\what{\Sym}^\ast F$ are homotopy factorization algebras.
A key example to bear in mind is the following. Let $E$ be a vector bundle on the manifold $M$, and let $\mathcal{E}$ denote the sheaf of smooth sections of $E$. We view $\mathcal{E}$ as the space of fields for some quantum field theory. The distributions that pair with these smooth sections form a cosheaf, as they are compactly supported. Denote this cosheaf by $\mathcal{E}^{\vee}$. Then, the results say that the prefactorization algebra sending
$$
U \to \Sym^\ast \E(U)^\vee = \oplus_k \Hom ( \E(U)^{\otimes k}, \R )_{S_k}
$$
is a factorization algebra; as is the completed symmetric algebra on $\E^\vee$.
If $E$ is a graded vector bundle, and the sheaf $\E$ of sections of $E$ is equipped with a differential which is a differential operator, then the (completed) symmetric algebra on the cosheaf of cochain complexes $\E^\vee$ is a factorization algebra and a homotopy factorization algebra. Indeed, in this case, the sheaf of cochain complexes given by sections of $E$ is a homotopy sheaf, so that the dual cosheaf of cochain complexes is a homotopy cosheaf. This is the only example of this result we will need for the rest of the paper.
//Proof://
Let us first prove the strict (non-homotopy) version of the result. Both $\Sym^\ast F$ and the completed symmetric algebra $\what{\Sym}^\ast F$ are prefactorization algebras, in an evident way. Thus, we need only verify the [[locality axiom | factorization algebra]].
Let $U$ be an open set in $M$ and $\mathfrak{U} = \{U_i \mid i \in I\}$ a factorizing cover of $U$. We need to prove the following three statements.
* $ \Sym^\ast F(U)$ is the cokernel of the map
$$
\oplus_{\alpha,\beta \in PI} \otimes_{i \in \alpha, j \in \beta} \Sym^\ast F ( U_{i,j} ) \to \oplus_{\gamma \in PI} \otimes_{k \in \gamma} \Sym^ast F( U_k ) .
$$
In this expression, $P I$ denotes, as [[before | factorization algebra]] the set of finite subsets of $I$, where for each $\alpha \in P I$, and every $i,j \in \alpha$, $U_i \cap U_j = \emptyset$.
The first thing to observe is the following. Since $F$ is a cosheaf,
\[
F(U_{i_1}) \oplus \cdots \oplus F(U_{i_n}) \cong F(U_{i_1} \cup \cdots \cup U_{i_n})
\]
if $U_{i_1}, \ldots, U_{i_n}$ are pairwise disjoint. Hence
$$
\Sym^\ast(F(U_{i_1})) \otimes \cdots \otimes \Sym^\ast(F(U_{i_n})) \cong \Sym^\ast(F(U_{i_1} \cup \cdots \cup U_{i_n})).
$$
Because we are using completed tensor products of nuclear spaces, the same is true for the completed symmetric algebra:
$$
\what{\Sym}^\ast (F(U_{i_1})) \otimes \cdots \otimes \what{\Sym}^\ast(F(U_{i_n})) \cong \what{\Sym}^\ast(F(U_{i_1} \cup \cdots \cup U_{i_n})).
$$
For each $\alpha \in P I$, let $U_\alpha = \amalg_{i \in \alpha} U_i$. We can rewrite the locality axiom as saying that $\Sym^\ast F(U)$ is the cokernel of
$$
\oplus_{\alpha,\beta \in P I } \Sym^\ast F (U_{\alpha} \cap U_{\beta} ) \to \oplus_{\gamma \in PI} \Sym^\ast F ( U_\gamma ) .
$$
This map compatible with the decomposition of $\Sym^\ast F(U)$ into symmetric powers. Thus, it suffices to show that, for all $m$,
$$
\Sym^m F(U) =\op{coker} \left( \oplus_{\alpha,\beta \in PI } \Sym^m ( F(U_{\alpha} \cap U_{\beta} ) \to \oplus_{\gamma \in P I} \Sym^m F(U_\gamma) \right).
$$
Now, observe that
$$
F(U)^{\otimes m} = F^{\boxtimes m} (U^m)
$$
where $F^{\boxtimes m}$ is the cosheaf on $U^m$ obtained as the external product of $F$ with itself $m$ times.
Thus it is enough to show that
$$
F^{\boxtimes m} (U^m) = \op{coker} \left( \oplus_{\alpha,\beta \in PI } F^{\boxtimes m} \left( (U_{\alpha} \cap U_{\beta})^m \right) \to \oplus_{\gamma \in P I} F^{\boxtimes m} \left( U_\gamma^m \right) \right).
$$
Our cover $\mf{U}$ is a factorizing cover. This means that, for every finite set of points $x_1,\ldots, x_k \in M$ we can find disjoint open subsets $U_{i_1},\ldots, U_{i_k}$ in the cover $\mf{U}$ with $x_i \in U_{i_i}$. This implies that the subsets of $U^m$ of the form $(U_\alpha)^m$, where $\alpha \in PI$, cover $U^m$. Further,
$$
(U_\alpha)^m \cap (U_\beta)^m = (U_{\alpha} \cap U_{\beta})^m.
$$
The desired isomorphism now follows from the fact that $F^{\boxtimes m}$ is a cosheaf on $M^m$.
A similar argument shows that the symmetric algebra of a homotopy cosheaf is a homotopy factorization algebra. The argument applies with out any change to the completed symmetric algebras, as we are working with pro-cochain complexes.
<<getTiddlerPassword>>
The goal of this section is to describe a natural class of factorization algebras, a class that includes all the factorization algebras we use in our theorems. In particular, we will show that given a nice cosheaf of vector spaces $F$, the functor $\Sym\, F: U \mapsto \Sym(F(U))$ is a factorization algebra. It is clear how this functor is a prefactorization algebra; the hard part is verifying when it satisfies the gluing axiom. We will pin down properties of a cosheaf that guarantee $\Sym\,F$ is a factorization algebra.
We begin by providing the definitions necessary to state the main result of this section. We then state the main result and explain its role for the rest of the wiki. Finally, we prove the lemmas that culminate in the proof of the main result.
!!! Preliminaries
Many of the definitions below are the obvious analogues for cosheaves of definitions in sheaf theory.
Throughout, let $M$ denote a locally compact, paracompact, Hausdorff space, and $R-mod$ the category of $R$-modules for a principal ideal domain $R$. Many definitions work more generally, of course.
''Definition''
A cosheaf $F: Open_M \rightarrow R-mod$ is //flabby// if every inclusion map $F(V \subset U): F(V) \rightarrow F(U)$ is a monomorphism.
''Definition''
Let $F$ be a flabby cosheaf. The //support// of a section $s \in F(U)$, denoted $|s|$, is the intersection of all open sets $V \subset U$ such that $s$ is in the image of the inclusion $F(V \subset U): F(V) \rightarrow F(U)$.
We will need to decompose any section into a sum of sections whose supports we control. In other words, we need partitions of unity.
''Definition''
Let $F$ be a flabby cosheaf on $M$. Given a locally finite open cover $\mathfrak{U} = \{U_\alpha\}$ of $M$, a //partition of unity// subordinate to the cover $\mathfrak{U}$ is a collection of endomorphisms $\{h_\alpha \in \Hom(F,F)\}$ such that
* for each $\alpha$, $|h_\alpha s| \subset U_\alpha$ for every $s \in F(M)$, and
* $\sum_\alpha h_\alpha = 1 \in \Hom(F,F)$.
In summary, we can decompose any global section $s$ as a sum of sections $\sum_\alpha h_\alpha s$ with support subordinate to our cover.
We say a cosheaf is //fine// if it admits a partition of unity for every locally finite open cover. The cosheaf of compactly-supported smooth functions (or distributions) on a manifold is a good example of a flabby, fine cosheaf. In general, a soft sheaf that admits partitions of unity yields a flabby, fine cosheaf by taking compactly-supported sections (cf. proposition 1.6 in ch. V of \cite{Bredon}).
Finally, here is our basic construction.
''Definition'' For $F$ a cosheaf, let $\Sym\,F$ denote the prefactorization algebra that assigns to any open set $U$ the module $\Sym(F(U))$.
!!! The main result
''Proposition'' For $F$ a flabby, fine cosheaf, $\Sym\,F$ is a factorization algebra taking values in the symmetric monoidal category $(R-mod, \otimes_R)$.
Thanks to this proposition, we obtain a large class of factorization algebras. Moreover, it contains exactly the factorization algebras we need. Let $E$ be a vector bundle on the manifold $M$, and let $\mathcal{E}$ denote the sheaf of smooth sections of $E$. We view $\mathcal{E}$ as the space of fields for some quantum field theory. The distributions that pair with these smooth sections form a cosheaf, as they are compactly supported. Denote this cosheaf by $\mathcal{E}^{\prime}$. The observables for this quantum field theory are the factorization algebra $\Sym\,\mathcal{E}^\prime$.
//Proof://
Since $\Sym\,F$ is a prefactorization algebra, we need only verify [[the gluing axiom | factorization algebra ]]. Let $U$ be an open set in $M$ and $\mathfrak{U} = \{U_\alpha\}$ a factorizing cover of $U$. Note that since $F$ is a cosheaf,
\[
F(U_{\alpha_1}) \oplus \cdots \oplus F(U_{\alpha_n}) \cong F(U_{\alpha_1} \cup \cdots \cup U_{\alpha_n})
\]
if $U_{\alpha_1}, \ldots, U_{\alpha_n}$ are pairwise disjoint. Hence
\[
\Sym(F(U_{\alpha_1})) \otimes \cdots \otimes \Sym(F(U_{\alpha_n})) \cong \Sym(F(U_{\alpha_1} \cup \cdots \cup U_{\alpha_n})).
\]
Thus, we can write the colimit appearing in the gluing axiom as
\[
\operatorname{colim} \Sym(F(V))
\]
where the colimit is taken over every open set $V \subset U$ obtained by taking a finite union of pairwise disjoint opens $U_{\alpha_1}, \ldots, U_{\alpha_n}$ from $\mathfrak U$. The maps arise from inclusions $V \subset V'$ of such opens.
Observe that we get an injection $\Sym(F(V)) \hookrightarrow \Sym(F(U))$ for every such $V$ because of the injection $F(V) \hookrightarrow F(U)$. (Here is where we use the hypothesis that $F$ is flabby.) Thus we get an injection
\[
\operatorname{colim} \Sym(F(V)) \hookrightarrow \Sym(F(U)).
\]
We need to show this map is also surjective. (This depends on $F$ being fine.) Once we've shown the map is an isomorphism, we have verified the gluing axiom.
Note that in the colimit above, all the maps respect the natural grading on symmetric algebras. Hence, we merely need to show we have an isomorphism
$$
\operatorname{colim} \Sym^k(F(V)) \rightarrow \Sym^k(F(U))
$$
for every $k$. We will now prove that every element of $\Sym^k(F(U))$ lives in $\Sym^k(F(V))$ for some $V$ in our diagram category.
We give a proof by induction on $k$. The $k = 0$ case is straightforward. The $k = 1$ case is a less obvious. We need the following lemma (\cite{Bredon}, Proposition 1.5 in ch. V).
''Lemma'' Let $F$ be a flabby cosheaf, and let $s \in F(U)$. Then
* the support $|s|$ is compact;
* for an open subset $V \subset U$, $|s| \subset V \Leftrightarrow s \in im (F(V \subset U): F(V) \rightarrow F(U))$;
* $|s| = \emptyset \Leftrightarrow s = 0$.
Pick a locally finite refinement $\mathfrak{V} = \{V_{\beta} \}$ of our cover $\mathfrak{U}$ and a partition of unity $\{h_{\beta} \}$ subordinate to $\mathfrak{V}$. Given any section $s \in F(U)$, it has compact support and hence vanishes on all but finitely many $V_\beta$. Then $s$ is the sum of finitely many terms $h_\beta s$, and we can view each $h_\beta s$ as a section of $F(V_\beta)$ by the lemma above. Each $V_\beta$ is contained in some $U_\alpha$, so $s$ arises from the colimit for $k =1$.
We need to prove the induction step. Suppose we have the isomorphism for $k=n-1$. Any element in $\Sym^n(F(U))$ is a sum of elements of the form $s_1 \otimes \cdots \otimes s_n$, so we prove the result for pure elements. By hypothesis, we can express $s_2 \otimes \cdots \otimes s_n$ as a sum of elements that each live in $\Sym^{n-1}(F(V))$ for some $V$ in our diagram category. Thus, we may reduce to the case where $s_2 \otimes \cdots \otimes s_n$ lives in such an open $V$. The section $s_1$ vanishes on all but finitely many $V_\beta$ in our locally finite cover $\mathfrak{V}$. Consider the intersection $|h_\beta s_1| \cap V$. Each connected component of this intersection must lie in some $U_\alpha$ as $V$ has this property. Hence $s_1 \otimes \cdots \otimes s_n$ is a sum of elements for which the support is in our diagram category. $\square$
Finally, we will see that $\ObsHomotopy$ defines not just a factorization algebra, but a prefactorization algebra, in $BD$ algebras; and that further, this factorization algebra is a quantization of the $P_0$ factorization algebra of classical observables. In addition, this factorization algebras is independent (up to homotopy) of the family $\{\Phi_t \mid t \in (0,\infty)\}$ of parametrices we used in the definition.
''Lemma''. //$\ObsHomotopy$ is a factorization algebra.//
''Proof.'' There is a filtration on the prefactorization algebra $\ObsHomotopy$ where the associated graded is the differential graded prefactorization algebra
$$
\begin{split}
\left(\op{Gr} \ObsHomotopy \right) (U) &= \Oo (\E(U)) \otimes \A [[\hbar]] \\
\op{Gr} \what{Q} &= Q.
\end{split}
$$
In other words, the differential on the associated graded just comes from the linear differential $Q : \E(U) \to \E(U)$.
Since the prefactorization algebra defined by $(\Oo(\E(U)), Q)$ is a factorization algebra, the same holds for $\ObsHomotopy$.
$\square$
''Lemma.'' //There is a quasi-isomorphism of $P_0$ factorization algebras//
$$
\ObsHomotopy \mod \hbar \simeq \Obs^{cl}.
$$
''Proof.''
(To come -- not hard...)
$\square$
The final thing we need to verify is that $\ObsHomotopy$ is independent (up to homotopy) of any choices required in the construction. The only choice we made was that of a family $\Phi_t$ of parametrices for the elliptic operator $[Q,\GF]$ on $\E$, depending smoothly on $t \in (0,1)$. We required that $\Phi_t$ is supported within $t$ of the diagonal.
''Lemma.'' //The factorization algebra $\ObsHomotopy$ is independent, up to homotopy, of the choice of family $\Phi_t$ of parametrices.//
''Proof.'' If $\Phi_t, \Psi_t$ are two such families, then they are canonically homotopic, with homotopy defined by
$$
F(s,t) = s \Phi_t + (1-s) \Psi_t.
$$
By construct the factorization algebra using the two-parameter family of parametrices $F(s,t)$, where $s \in [0,1]$ and $t \in (0,1)$, we find a factorization algebra over $\Omega^\ast( [0,1]) \otimes \A$, interpolating between that constructed from $\Phi$ and that constructed from $\Psi$.
$\square$
Physicists normally think of a classical field theory as being associated to an action functional. In this section we will show how to construct a classical field theory in our sense from an action functional.
We will work in a quite general setting. [[Recall | elliptic moduli problems]] that we defined a local $L_\infty$ algebra on a manifold $M$ to be a sheaf of $L_\infty$ algebras where the structure maps are given by differential operators. We will think of a local $L_\infty$ algebra $\mscr{L}$ on $M$ as defining a formal moduli problem cut out by some elliptic equations. We will use the notation $B \mscr{L}$ to denote this formal moduli problem.
We want to take the derived critical locus of a local action functional
$$
S \in \Ool ( B \mscr{L} )
$$
of cohomological degree $0$. (We also need to assume that $S$ is at least quadratic: this means that the base-point of our formal moduli problem $B \mscr{L}$ is a critical point of $S$). We have [[seen |The exterior derivative of a local action functional]] how to apply the exterior derivative to a local action functional $S$ yields an element
$$
\d S \in C^\ast_{loc} ( \L , \L^! [-1])
$$
which we think of as being a local one-form on $B \mscr{L}$.
The critical locus of $S$ is the zero locus of $\d S$. We thus need to explain how to construct a new local $L_\infty$ algebra which we interpret as being the zero locus of $\d S$.
!!! Finite dimensional model
We will first describe the analogous construction in finite dimensions. Let $\g$ be an $L_\infty$ algebra, $M$ be a $\g$-module of finite total dimension, and $\alpha$ be a closed degree zero element of $C^\ast_{red}(\g, M)$. The subscript $red$ indicates that we are taking the reduced cochain complex, so that $\alpha$ is in the kernel of the augmentation map $C^\ast(\g, M) \to M$.
We should think of $M$ as a dg vector bundle on the formal derived moduli problem $B \g$, and $\alpha$ as a section of this vector bundle. The condition that $\alpha$ is in the reduced cochain complex means translates into the statement that $\alpha$ vanishes at the basepoint of $B \g$. We are interested in constructing the $L_\infty$ algebra representing the zero locus of $\alpha$.
The commutative dga representing this zero locus is given by the total complex of the double complex
$$
\dots \to C^\ast(\g, \wedge^2 M^\vee ) \xto{\alpha^\vee }C^\ast(\g,M^\vee) \xto{\alpha^\vee} C^\ast(\g).
$$
This commutative dga is the symmetric algebra on the dual of $\g[1] \oplus M[-1]$. It follows that this commutative dga is the ~Chevalley-Eilenberg cochain complex of $\g \oplus M[-2]$, equipped with a certain $L_\infty$ structure.
Note that $\g \oplus M[-2]$ has a natural semi-direct product $L_\infty$ structure, arising from the $L_\infty$ structure on $\g$ and the $\g$ action on $M[-2]$. This $L_\infty$ structure corresponds to the case $\alpha = 0$.
''Lemma.''
//The $L_\infty$ structure on $\g \oplus M[-2]$ describing the zero locus of $\alpha$ is a deformation of the semidirect product $L_\infty$ structure, obtained by adding to the structure maps $l_n$ the maps//
$$
\begin{split}
D_n \alpha : \g^{\otimes n} & \to M \\
X_1 \otimes \dots \otimes X_n & \mapsto \frac{\partial}{\partial X_1} \dots \frac{\partial}{\partial X_n} \alpha .
\end{split}
$$
''Proof.''
The proof is a straightforward computation.
$\square$
Note that the maps $D_n \alpha$ in the statement of the lemma are simply the homogeneous components of the cochain $\alpha$.
We will let $Z(\alpha)$ denote $\g \oplus M[-2]$, equipped with this $L_\infty$ structure.
Recall that the formal moduli problem $B \g$ is the functor from dg Artin rings $(R,m)$ to simplicial sets, sending $(R,m)$ to the simplicial set of ~Maurer-Cartan elements of $\g \otimes m$. In order to check that we have constructed the correct derived zero locus for $\alpha$, we should describe the formal moduli problem associated $Z(\alpha)$.
Thus, let $(R,m)$ be a dg Artin ring, and $x \in \g \otimes m$ be an element of degree $1$, and $y \in M \oplus m$ be an element of degree $-1$. Then, $(x,y)$ satisfies the ~Maurer-Cartan equation in $Z(\alpha)$ if and only if:
# $x$ satisfies the ~Maurer-Cartan equation in $\g \otimes m$.
# $\alpha(x) = \d_x y \in M$, where $\d_x : M \to M$ is the differential obtained by deforming the original differential by that arising from the ~Maurer-Cartan element $x$.
In other words, we see that an $R$-point of $B Z(\alpha)$ is an $R$-point $x$ of $B \g$, and a homotopy between $\alpha(x)$ and $0$, in the fibre $M_x$ of the bundle $M$ at $x \in B \g$.
!!! The derived critical locus of a local functional
Let us now return to the situation where $\L$ is a local $L_\infty$ algebra on a manifold $M$, and $S \in \Oo (B \L)$ is a local functional which is at least quadratic. We let
$$
\d S \in C^\ast_{loc}(\L, \L^! [-1] )
$$
denote the exterior derivative of $S$. Note that $\d S$ is in the reduced cochain complex (that is, the kernel of the augmentation map $C^\ast_{loc}(\L,\L^![-1] ) \to L^![-1]$).
Let
$$
\d_n S : \L^{\otimes n} \to \L^!
$$
be the $n^{th}$ Taylor component of $\d S$. The fact that $\d S$ is a local cochain means that $\d_n S$ is a polydifferential operator.
''Definition.''
//The derived critical locus of $S$ is the local $L_\infty$ algebra obtained by adding to the structure maps $l_n$ of the semi-direct product $L_\infty$ algebra $\L \oplus \L^![-3]$ the maps//
$$
\d_n S : \L^{\otimes n} \to \L^!.
$$
Let us denote this local $L_\infty$ algebra by $\op{Crit}(S)$. If $(R,m)$ is an auxiliary Artinian dg ring, then a solution to the ~Maurer-Cartan equation in $\op{Crit}(S) \otimes m$ consists of the following data.
# A ~Maurer-Cartan element $x \in \L \otimes m$.
# An element $y \in \L^! \otimes m$, such that
$$
(\d S) (x) = \d_x y.
$$
Here, $\d_x y$ is the differential on $L^! \otimes m$ induced by the ~Maurer-Cartan element $x$. These two equations say that $x$ is an $R$-point of $B \L$ which satisfies the ~Euler-Lagrange equations up to a homotopy specified by $y$.
!!! Symplectic structure on the derived critical locus
Recall that a classical field theory is given by a local $L_\infty$ algebra which is elliptic, and which also has an invariant pairing of degree $-3$. The pairing on the local $L_\infty$ algebra $\op{Crit}(S)$ constructed above is evident: it is given by the natural bundle isomorphism
$$
(L \oplus L^![-3] )^! [-3] \iso L \oplus L^![-3].
$$
In other words, the pairing arises from the bundle map
$$
L \otimes L^! \to \op{Dens}_M.
$$
''Lemma''.
//This pairing is invariant under the $L_\infty$ structure constructed from $\d S$.//
''Proof.''
The original $L_\infty$ structure on $\L \oplus \L^![-3]$ (that is, the $L_\infty$ structure not involving $S$) is easily seen to be invariant. We will verify that the deformation of this structure coming from $S$ is also invariant.
We need to show that, if
$$
\alpha_1, \dots,\alpha_{n+1} \in \L_c \oplus \L^!_c [-3]
$$
are compactly supported sections of $L \oplus L^1[-3]$, that
$$
\ip{l_n ( \alpha_1,\dots, \alpha_{n} ) ,\alpha_{n+1} }
$$
is totally antisymmetric in the variables $\alpha_i$. Now, the part of this expression which comes from $S$ is just
$$
\left( \frac{\partial}{\partial \alpha_1} \dots \frac{\partial}{\partial \alpha_{n+1}} \right) S (0).
$$
The fact that partial derivatives commute, and the shift in grading coming from the fact that $C^\ast(\L_c) = \Oo( \L_c[1])$, immediately implies that this is totally antisymmetric.
$\square$.
Note that, although the local $L_\infty$ algebra $\op{Crit}(S)$ has a symplectic form, it does not always define a classical field theory. It only does so under the additional assumption that the local $L_\infty$ algebra $\op{Crit}(S)$ is elliptic.
Before we discuss the concepts specific to classical field theory, we will explain some general techniques from deformation theory.
In ordinary algebraic geometry, the fundamental objects are commutative rings. In derived algebraic geometry, commutative rings are replaced by commutative differential graded rings concentrated in non-positive degrees (or, if one prefers, simplicial commutative rings; over $\Q$, there is no difference).
We are interested in formal derived geometry, which is described by nilpotent commutative dgas.
''Definition.''
//An Artinian dga over a field $K$ of characteristic zero is a differential graded $K$-algebra $R$, concentrated in degrees $\le 0$, such that//
# //Each graded component $R^i$ is finite dimensional, and $R^i = 0$ for $i << 0 $//.
# //$R$ has a unique maximal differential ideal $m$ such that $R / m = K$, and such that $m^N = 0$ for $N >> 0$.//
Given the first condition, the second condition is equivalent to the statement that $H^0(R)$ is Artinian in the classical sense.
The category of Artinian dg rings is a simplicially enriched category. A map $R \to S$ is simply a map of dg rings taking the maximal ideal $m_R$ to that of $m_S$. Equivalently, such a map is a map of non-unital dg rings $m_R \to m_S$. An $n$-simplex in the space $\op{Hom}(R,S)$ of maps from $R$ to $S$ is defined to be a map of non-unital dg rings
$$
m_R \to m_S \otimes \Omega^\ast(\bigtriangleup^n)
$$
where $\Omega^\ast(\bigtriangleup^n)$ is some commutative algebra model for the cochains on the $n$-simplex. (Normally, we will work over $\R$, and $\Omega^\ast(\bigtriangleup^n)$ will be the usual de Rham complex).
We will (temporarily) let $\op{Art}$ denote the simplicially enriched category of Artinian dg rings over $K$.
''Definition.''
//A formal moduli problem over a field $k$ is a functor (of simplicially enriched categories)//
$$
F : \op{Art}_k \to \op{sSets}
$$
//from $\op{Art}_k$ to the category $\op{sSets}$ of simplicial sets, with the following additional properties.//
# //$F(k)$ is contractible.//
# //$F$ takes surjective maps of dg Artinian rings to fibrations of simplicial sets.//
# //Suppose that $A,B,C$ are dg Artinian rings, and that $B \to A$, $C \to A$ are surjective maps. Then we can form the fibre product $B \times_A C$. We require that the natural map//
$$
F (B \times_A C) \to F(B) \times_{F(A)} F(C)
$$
//is a weak homotopy equivalence.//
Note that, in light of the second property, the fibre product $F(B) \times_{F(A)} F(C) $ coincides with the homotopy fibre product.
The category of formal moduli problems is itself simplicially enriched, in an evident way. If $F, G$ are formal moduli problems, and $\phi : F \to G$ is a map, we say that $\phi$ is a weak equivalence if for all dg Artinian rings $R$, the map
$$
\phi(R) : F(R) \to G(R)
$$
is a weak homotopy equivalence of simplicial sets.
!!! Formal moduli problems and $L_\infty$ algebras
One very important way in which formal moduli problems arise is as the solutions to the ~Maurer-Cartan equation in an $L_\infty$ algebra. As we will see later, all formal moduli problems are equivalent to formal moduli problems of this form.
If $\g$ is an $L_\infty$ algebra, and $(R,m)$ is a dg Artinian ring, we will let
$$
\op{MC} ( \g \otimes m)
$$
denote the simplicial set of solutions to the ~Maurer-Cartan equation in $\g \otimes m$. Thus, an $n$-simplex in this simplicial set is an element
$$
\alpha \in \g \otimes m \otimes \Omega^\ast(\bigtriangleup^n)
$$
of cohomological degree $1$, which satisfies the ~Maurer-Cartan equation
$$
\d \alpha + \sum_{n \ge 2} \tfrac{1}{n!} l_n ( \alpha , \dots ,\alpha) = 0.
$$
It is a standard lemma that sending $R$ to $\op{MC}(\g \otimes m)$ defines a formal moduli problem. We will often use the notation $B \g$ to denote this formal moduli problem.
If $\g$ is finite dimensional, then a ~Maurer-Cartan element of $\g \otimes m$ is the same thing as a map of commutative dgas
$$
C^\ast (\g) \to R
$$
which takes the maximal ideal of $C^\ast(\g)$ to that of $R$.
Thus, we can think of the ~Chevalley-Eilenberg cochain complex $C^\ast(\g)$ as the algebra of functions on $B \g$.
Under the dictionary between formal moduli problems and $L_\infty$ algebras, a dg vector bundle on $B \g$ is the same thing as a dg module over $\g$. The cotangent complex to $B \g$ corresponds to the $\g$-module $\g^\vee[-1]$. The tangent complex corresponds to the $\g$-module $\g[1]$.
If $M$ is a $\g$-module, then sections of the corresponding vector bundle on $B \g$ is the Chevalley cochains with coefficients in $M$. Thus, we can define $\Omega^1( B\g)$ to be
$$
\Omega^1(B \g ) = C^\ast (\g, \g^\vee[-1] ) .
$$
Similarly, the complex of vector fields on $B \g$ is
$$
\op{Vect}( B \g) = C^\ast(\g, \g[1] ) .
$$
Note that, if $\g$ is finite dimensional, this is the same as the cochain complex of derivations of $C^\ast(\g)$. Even if $\g$ is not finite dimensional, this the complex $\op{Vect}(B \g)$ is, up to a shift of one, the Lie algebra controlling deformations of the $L_\infty$ structure on $\g$.
!!! The fundamental theorem of deformation theory
The following statement is at the heart of the philosophy of deformation theory:
<<<
There is an equivalence of $(\infty,1)$ categories between the category of differential graded Lie algebras, and the category of formal pointed derived moduli problems.
<<<
In a different guise, this statement goes back to Quillen's work ([[Qui69]]) on rational homotopy theory. A precise formulation of this theorem has been proved by Hinich ([[Hin01]]); more general theorems of this nature are considered in ([[Lur10]]), which is also an excellent survey of these ideas.
It would take us too far afield to describe the language in which this statement can be made precise. We will simply use this statement as motivation: we will only consider formal moduli problems described by $L_\infty$ algebras, and this statement asserts that we lose no information in doing so.
The factorization algebra of quantum observables is particularly easy to construct for a free theory, because it is well-defined at length scale 0, as we show in this section. Below, we explain this construction, and in subsequent sections, we work through standard examples of free theories from physics, such as the [[free particle | Weyl algebra redux]] moving in a Riemannian manifold and [[the free holomorphic boson]], which recovers the Heisenberg vertex algebra.
!!! Quantum observables at scale 0
Consider a [[free BV theory|BV theory]] on a smooth manifold $M$ whose space of fields is $\E$, whose symplectic pairing is $\langle -,-\rangle$, whose action functional is obtained from the differential operator $Q$ , and whose gauge-fixing operator is $Q^*$. The cochain complex of fields $(\E,Q)$ is elliptic and describes the derived critical locus of the action functional $S(\phi) = \langle \phi, Q\phi \rangle$. On any open set $U \subset M$, the classical observables supported in $U$ is the commutative dga
\[
Obs^{cl}(U) = (\csym(\E^\vee(U)), Q),
\]
where $\E^\vee(U)$ denotes the distributions dual to $\E$ with compact support in $U$ and $Q$ is the derivation given by extending the natural action of $Q$ on the distributions.
Ideally, we could define the quantum observables at scale 0 as
\[
Obs^q(U) = (\Sym(\E^\vee(U)) \otimes \R[[\hbar]], Q + \hbar \Delta_0),
\]
where $\Delta_0 = - \partial_K$, with $K$ the kernel corresponding to the symplectic pairing. Since $K$ is distributional, though, we run into the usual problems with pairing distributions. Thus, this putative cochain complex is ill-defined.
For a free field, we can remedy the situation by working with ``smeared" observables. Consider the inclusion of cochain complexes
\[
(\E, Q) \hookrightarrow (\bar{\E}, Q),
\]
where $\bar{\E}$ denotes the distributional completion of $\E$. [[Elliptic regularity|Atiyah-Bott lemma]] insures that this inclusion is in fact a quasi-isomorphism. For the same reason, we can replace the classical observables with smooth observables
\[
\widetilde{Obs}^{cl}(U) = (\Sym(\bar{\E}^\vee(U)), Q) \overset{\cong}{\hookrightarrow} Obs^{cl}(U).
\]
The BV Laplacian $\Delta_0$ is well-defined on these smeared observables, so we can construct the quantum observables
\[
\widetilde{Obs}^q(U) = (\Sym(\bar{\E}^\vee(U)) \otimes \R[[\hbar]], Q + \hbar \Delta_0).
\]
This construction is substantially easier than our construction of the observables for an interacting field.
//Remark:// Many of the simplest observables, like a delta function, are distributional. Thus, to compute its expectation value, we have to find a quasi-isomorphic ``smeared" observable.
!!! Gauge theories
We often want to study gauge theories, where the space of fields has some redundancy. More precisely, the action functional is invariant under some the action of some Lie group $G$ on the space of fields. (In the perturbative formalism, we consider situations where the action functional is invariant under some Lie algebra action.) Hence we actually wish to work with the quotient $\Fields/G$, as the redundancy is then eliminated. Unless the action of $G$ is free, the naive quotient is ill-behaved (and usually does not exist in the framework of ordinary geometry). Instead, following the philosophy of derived geometry, we work with the derived quotient.
In the perturbative setting, where a Lie algebra $\mathfrak{g}$ is acting on $\Fields$, the derived quotient is quite easy to describe. We replace $\Fields$ by the dg manifold whose functions are the cochain complex $\mathcal{O}(\mathfrak{g}[1] \oplus \Fields)$ with the differential encoding how $\mathfrak{g}$ acts on $\Fields$.We then apply the derived critical locus construction to this dg manifold, as before. For a more thorough discussion, see chapter 5 of \cite{webbook}.
!!! Nonlinear sigma models
IMPROVE: we need to actually describe these (formal nbhd of fixed classical solution, rather than pullback of tangent bundle, etc)
Our formalism only applies for fields that are sections of vector bundles, because it is perturbative in approach. The philosophy is that we are doing perturbative computations around some fixed classical solution to the ~Euler-Lagrange equations. These computations incorporate the "quantum corrections" that arise from infinitesimal perturbations around that classical solution. Thus this philosophy indicates how to treat situations where the fields are maps from $M$ into some other manifold $N$, or sections of some complicated fiber bundle over $N$. We describe our approach where $\Fields = Maps(M,N)$. First, pick a classical solution $\phi$ to the ~Euler-Lagrange equations for this nonlinear sigma model. "Infinitesimal perturbations" of $\phi$ ought to stay quite close to $\phi$, so we suppose that we can restrict attention to maps whose images stay in some small tubular neighborhood of $\phi(M) \subset N$. In other words, we work with sections of the vector bundle $\phi^\ast TN$ over $M$. Now apply the BV formalism from above.
When considering non-linear sigma models, however, one has to ensure that the quantization is independent of the choice of linearization of the space of fields near a given solution. This can be analyzed in our framework by working relative to an appropriate base ring.
A careful treatment of a particular non-linear sigma model is presented in \cite{elliptic}.
Ezra Getzler, //Lie theory for nilpotent $L_\infty$ algebras// math.AT/0404003 (2004)
Let
$$
\A = \colim_{\delta \to 0} \Omega^\ast( (0,\delta)) .
$$
Because $\A$ is the colimit of a countable family of nuclear spaces, it is again a nuclear space. However, $\A$ is //not// Hausdorff.
Let
$$
\ObsHomotopy(M) = \Oo(\E) \otimes \A[[\hbar]].
$$
This is the space of global sections of the factorization algebra.
Because the tensor product of nuclear spaces commutes with limits, we can identify
$$
\ObsHomotopy(M) = \left\{\prod_{n \ge 0} \left( \Hom(\E^{\otimes n}, \R)_{S_n} \otimes \A \right)\right\} [[\hbar]]
$$
Further, since the $\E^{\otimes n}$ is a nuclear Frechet space, and since tensor product with the dual of a nuclear Frechet space commutes with countable colimits,
$$
\Hom(\E^{\otimes n}, \R) \otimes \A = \colim_{\delta} \Hom(\E^{\otimes n}, \R) \otimes \Omega^\ast((0,\delta) ) .
$$
Thus, if $O \in \ObsHomotopy(M)$, we can expand $O$ as a sum
$$
O = \sum_{i,k} \hbar^i O_{i,k}
$$
where
$$
O_{i,k} \in \colim_{\delta \to 0} \Hom ( \E^{\otimes k}, \Omega^\ast((0,\delta)) _{S_k}.
$$
!!! The Batalin-Vilkovisky operator on global observables
So far we have defined global observables just as a cochain complex. Next we will define the differential on $\ObsHomotopy(M)$.
Let $\Phi_t$ be the parametrix associated to $t \in (0,\infty)$. Let
$$
\Psi_t = \frac{\d}{\d t} \Phi_t.
$$
Since the difference between any two parametrices is smooth, the derivative $\Psi_t$ is an element
$$\Psi_t \in \cinfty((0,\infty)) \otimes \E(M) \otimes \E(M).$$
Further, $\Psi_t$ has proper support.
Recall that the propagator $P(\Phi_t)$ associated to $\Phi_t$ is defined by
$$
P(\Phi_t) = (\GF \otimes 1) \Phi_t.
$$
Let
$$
P(\Psi_t) = (\GF \otimes 1) \Psi_t.
$$
Contraction with $P(\Psi_t)$ defines a linear operator
$$
\partial_{P(\Psi_t)} : \Oo(\E) \to \Oo(\E).
$$
This operator will be used to define an infinitesimal form of the renormalization group flow.
Recall that, associated to the propagator $\Phi_t$ is a BV operator $\Delta_{\Phi_t}$ defined by contraction with the tensor
$$
([Q,\GF] \otimes 1) \Phi_t - K_0.
$$
Both the operators $\Delta_{\Phi_t}$ and $\delta_{P(\Psi_t)}$ depend smoothly on $t \in (0,\infty)$, and both can thus be viewed as $\cinfty((0,\infty))$-linear operators on $\Oo(\E)$.
We will let
$$
\Delta = \Delta_{\Phi_t} + \d t \partial_{P(\Psi_t)} : \Oo(\E) \otimes \Omega^\ast( (0,\infty) ) \to \Oo(\E) \otimes \Omega^\ast( (0,\infty) ).
$$
This operator descends to an operator
$$
\Delta : \Oo(\E) \otimes \A \to \Oo(\E) \otimes \A.
$$
''Lemma''. $\Delta^2 = 0$//, and//
$$
[Q + \d_{dR} , \Delta ] = 0.
$$
//(Here $Q : \Oo(\E) \to \Oo(\E)$ is the linearized differential on $\Oo(\E)$, and $\d_{dR}$ is the de Rham differential on $\Omega^\ast((0,\infty))$).//
''Proof.'' This follows immediately from the definitions.
$\square$
Thus, $Q + \d_{dR} + \hbar \Delta$ gives
$$\ObsHomotopy(M) = \Oo(\E(M)) \otimes \A [[\hbar]]$$
the structure of BD algebra, linear over $\A$. Let $\{-,-\}$ be the associated bracket, defined by the formula
$$
\{\alpha,\beta\} = \Delta(\alpha \beta) + (\Delta \alpha) \beta \pm \alpha \Delta \beta.
$$
!!! The effective interaction
For each $t \in (0,\infty)$, we have an element
$$
I[\Phi_t] \in \Oo(\E)[[\hbar]] .
$$
This depends smoothly on $t \in (0,\infty)$, and so defines an element
$$
I \in \Oo(\E) \otimes \cinfty((0,\infty)) [[\hbar]].
$$
Using the natural map $\cinfty( (0,\infty)) \to \A$ we get an element
$$ I \in \ObsHomotopy(M) = \Oo(\E) \otimes \Omega^\ast(\Px_0)[[\hbar]].$$
//Remark:// In the case $M$ is not compact, there are some small subtleties which I should clarify some time... Basically $I$ does not have compact support, but has proper support, which is enough to define the bracket.
Let us define the quantized differential $\what{Q}$ on $\Oo(\E) \otimes \Omega^\ast(\Px) [[\hbar]]$ by the formula
$$
\what{Q} = Q + \d_{dR} + \hbar \Delta + \{I,-\}.
$$
''Lemma''. $\what{Q}^2 = 0$.
''Proof.'' This follows immediately from the fact that $I[\Phi_t]$ satisfies the renormalization group equation and the quantum master equation. Let us expand the bracket $\{-,-\}$ as
$$
\{-,-\} = \{-,-\}_t^0 + \d t\{-,-\}_t^1.
$$
Both are operators on $\Oo(\E) \otimes \cinfty((0,\infty))$.
The quantum master equation for $I[\Phi_t]$ says that
$$
Q I[\Phi_t] + \hbar \Delta_{\Phi_t} I[\Phi_t] + \tfrac{1}{2} \{I[\Phi_t], I[\Phi_t] \}_t^0 = 0.
$$
The renormalization group equation says that
$$
\frac{\d}{\d t} I[\Phi_t] + \hbar \partial_{ P(\Psi_t) } + \tfrac{1}{2} \{ I[\Phi_t], I[\Phi_t] \}_t^1 = 0.
$$
$\square$
We will think of closed elements of the cochain complex $\ObsHomotopy(M)$, with the differential $\what{Q}$, as observables which satisfy the renormalization group equation up to homotopy.
Let $ M $ be a manifold. In this page we will, for simplicity, assume that $M$ is compact; this restriction will shortly be removed. Suppose we have a [[ quantum field theory]] on $ M $, in the sense of \cite{webbook}. Let us denote the space of fields of this quantum field theory by $ \mathcal{E} $. Our definition of quantum field theory assumes that the space $ \mathcal E $ of fields is the space of global sections of some $ \mathbb{Z} $-graded vector bundle $ E $ on $ M $.
In this page we will introduce the concept of observable of a quantum field theory. An observable for a quantum field theory is simply a first order deformation of the theory (which may not, however, satisfy all the locality properties of the original theory).
!!! Global observables
Let $ \{I[L]\} \in \mathcal O( \mathcal E ) [[\hbar]] $ denote the effective interactions which encode the quantum field theory.
''Definition.''
//A global observable of cohomological degree $ i $ is a collection of functionals//
$$
\{ O[L] \in \mathcal O( \mathcal E ) [[\hbar]] \mid L \in \mathbb{R}_{> 0} \},
$$
//which are of cohomological degree $ i $, and which are such that, if $ \delta $ is a parameter of cohomological degree $ -i $ and square zero, the collection of functionals//
$$
I[L] + \delta O[L] \in \mathcal O(\mathcal E) [[\hbar]] [\delta]/\delta^2
$$
//satisfies the renormalization group equation. We will let $ \operatorname{Obs}^i(M) $ denote the space of global observables of cohomological degree $ i $. //
Thus, the collection of effective interactions $ \{I[L] + \delta O[L]\} $ satisfy most of the axioms needed to define a family of quantum field theories over the base ring $ \mathbb{R}[\delta]/ \delta^2 $. The only axiom which is not satisfied is the locality axiom: we have not imposed any constraints on the behaviour of the $ O[L] $ as $ L \rightarrow 0 $.
''Lemma.''
//Suppose $ \{O[L]\} $ is a collection of functionals defining a global observable of cohomological degree $ i$. Define $\left\{ \left( \widehat{Q} O\right) [L] \right\} $ by//
$$
\left(\widehat{Q} O\right)[L] = Q O[L] + \{ I[L], O[L] \}_L + \hbar \Delta_L O[L].
$$
//Then, $\left\{\left(\widehat{Q} O\right)[L]\right\} $ defines an observable of cohomological degree $ i+1 $.//
//Further, sending//
$$
\{O[L]\} \rightarrow \left\{ \left( \widehat{Q} O\right) [L] \right\}
$$
//defines a differential on the graded vector space $\operatorname{Obs}_{strict}^\ast(M)$, making it into a cochain complex.//
''Proof''
The first step is to show that $\left\{ \left( \widehat{Q} O\right) [L] \right\}$ satisfies the renormalization group equation for observables of cohomological degree $i+1$. This follows from the compatability between the quantum master equation and the renormalization group equation described in \cite{webbook}, Chapter 5. The fact that the effective interactions $ I[L] $ satisfy the quantum master equation implies that $ \widehat{Q}^2 = 0 $.
$\square$
Note that observables which are closed for this differential are precisely first-order deformations of the family $I[L]$ of effiective interactions, which satisfy both the renormalization group equation and the quantum master equation.
!!!! Global observables using parametrices
''Lemma''
//To specify a global observable is the same as to specify a functional $ O[\Phi] \in \mathcal {O}( \mathcal{E} )$ for all parametrices $\Phi$, such that the renormalization group equation//
$$
I[\Phi] + \delta O[\Phi] = W ( P(\Phi) - P(\Psi), I[\Psi] + \delta O[\Psi] ) \operatorname{mod } \delta^2
$$
//holds.//
Using this definition of a global observable, the differential $ \widehat{Q} $ on $ \ObsStrict^\ast(M) $ is defined by
$$
\left( \widehat{Q} O \right) [\Phi] := Q O[\Phi] + \{ I[\Phi], O[\Phi] \}_{\Phi} + \hbar \Delta_{\Phi} I[\Phi] \in \mathcal{O}(\mathcal{E}) [[\hbar]]
$$
In what follows, we will always use this alternative (and equivalent) definition of global observable.
Grothendieck, A. //Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires.// Annales de l'institut Fourier, 4 (1952), p. 73-112
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Hinich, DG coalgebras as formal stacks, 2001
The next few pages will construct the factorization algebra of homotopy observables of a quantum field theory. This construction of a factorization algebra has the advantage that it is a BD factorization algebra, and provides a BD quantization of the $P_0$ factorization algebra of the classical field theory.
The disadvantage, however, is that this factorization algebra is only defined over a somewhat unpleasant (non-Hausdorff) topological dga $\A$.
!!! The base ring
''Definition''. //Let $\A$ denote the topological commutative differential graded algebra defined as the colimit//
$$
\A = \op{colim}_{\delta \to 0} \Omega^\ast ( 0,\delta )
$$
//of algebras of forms on the intervals $(0,\delta$). This colimit is taken in the category of all topological vector spaces, and not in the category of Hausdorff spaces.//
Thus, $\A$ is the algebra of germs of forms at $0$ on $(0,1)$. It is easy to verify that $H^\ast(\A) = \R$. However, the non-Hausdorff nature of $\A$ means that there are no continuous linear maps
$$
\A \to \R.
$$
In particular, there quasi-isomorphism $\R \to \A$ does not admit a homotopy inverse.
We will let
$$
\A_0 = \colim_{\delta \to 0} \Omega^\ast( [0,\delta) )
$$
denote the algebra of germs of forms at $0$ on $[0,1)$. Note that while $\A_0$ non-Hausdorff, there is, of course, a quasi-isomorphism
$$
\A_0 \to \R.
$$
!!! $\A$-modules.
Let $M$ be a nuclear space which is the dual of a nuclear Frechet space. We define
$$
M \otimes \A = \colim_{\delta \to 0} (M \otimes \Omega^\ast(0,\delta).
$$
''Definition'' //An $\A$-module $N$ is good if it can be written as a countable product of tensor products of the form $M \otimes \A$, where $M$ is the dual of a nuclear Frechet space.//
''Claim'' //Good $\A$-modules form a symmetric monoidal category under the completed projective tensor product...//
!!! The quantization theorem
Let us suppose we have a classical field theory on a manifold $M$, with associated $P_0$ factorization algebra $\Obs^{cl}$.
# //Every quantum field theory, in the sense of \cite{webbook}, yields a BD factorization algebra $\Obs_{\A}^q$ on $M$, in the category of good modules over $\A[[\hbar]]$.//
# //There exists a $P_0$ factorization algebra $\Obs6{cl}_{\A_0}$, over $\A_0$, together with quasi-isomorphisms of $P_0$ factorization algebras over $\A_0$,//
$$
\Obs_{\A}^q \otimes_{\A[[\hbar]] } \A \leftarrow \Obs^{cl}_{\A_0} \to \Obs^{cl}.
$$
//(where the $\A_0$-module structure on both side is induced by the maps $\A_0 \to \A$ and $\A_0 \to \R$).//
''Conjecture'' (Vague statement) //Every BD quantization of $\Obs^{cl}$ arises in this way//.
!!! Families of parametrices
The construction of $\ObsHomotopy$ will require the choice of a family of a smooth family $\Phi_t$ of parametrices, for $t \in (0,1)$ satisfying certain properties. However, the resulting factorization algebra is independent, up to coherent homotopy, of this choice. Thus, we will start by describing what it means to have a smooth family of parametrices, and what properties we require of our particular smooth family $\Phi_t$.
Let $\Px$ denote the set of parametrices. Let $X$ be an auxiliary manifold. We will first define what it means for a map $X \to \Px$ to be smooth.
''Definition.'' //A smooth map $f : X \to \Px$ is an element//
$$
\Phi \in \cinfty(X) \otimes \br{\E}(M) \otimes \br{\E}(M)
$$
//(where the tensor products, as always, are completed), such that for each $x \in X$, $\Phi_x$ is a parametrix.//
Now, let us choose a smooth map
$$
\Phi : (0,\infty) \to \Px
$$
of parametrices. We let $\Phi_t$ denote the value of this at $t \in (0,\infty)$. Let us assume that $\Phi_t$ is supported within $t$ of the diagonal in $M \times M$.
If $\Phi, \Phi' : (0,\infty) \to \Px$ are two such maps, then they are canonically homotopic. The homotopy is defined by
$$
\begin{split}
\Psi : (0,\infty) \times [0,1] &\to \Px \\
\Psi (t,s) = s \Phi_t + (1-s) \Phi'_t.
\end{split}
$$
!!! Outline of the construction
# [[Global homotopy observables]] In this page, we construct the global observables using the renormalization group flow up to homotopy.
# [[Prefactorization structure]] Here, we describe the local observables (those supported on some open subset), and explain how they form a BD prefactorization algebra.
# [[Factorization structure]] Finally, we verify that the BD prefactorization algebra of homotopy observables is actually a BD factorization algebra, and that it provides a quantization of the $P_0$ factorization algebra of classical observables.
!!! Local observables
So far, we have defined the cochain complex $\Obs^\ast(M)$ of [[global observables]] on the whole manifold $M$. If $U \subset M$ is an open subset of $M$, we would like to isolate those observables which are "supported on $U$".
We would like to say that an observable $\{O[L]\}$ is supported on an open set $U$ if, as $L \to 0$, the distributions in powers of $M$ which are the components of $O[L]$ become concentrated on powers of $U$.
It is difficult to make this idea into a workable definition. This is largely because the heat kernel $K_L$ is supported on the whole manifold $M \times M$, even if $L$ is very small. Because the renormalization group flow satisfied by the $O[L]$ involves the heat kernel, we would also expect that $O[L]$ is supported on the whole manifold, even if as $L \to 0$, the distribution $O[L]$ tends to zero away from the open set $U$.
In what follows, we will solve this problem by using an alternative way to access the small scale behaviour of a quantum field theory. The idea is to use a different version of the renormalization group flow, which uses an arbitrary parametrix for the operator $[Q,\GF]$ instead of the heat kernel. In this way, we will construct a factorization algebra $\Obs$ consisting of observables satisfying the parametrix version of the renormalization group flow.
!!! BD algebra structure
If we fix a scale $k \in (0,\infty)$, we get a map
$$
\op{ev}[k] : \Obs^\ast(M) \to \Oo(\E(M))[[\hbar]]
$$
defined by
$$
\op{ev}[k] \{O[L]\} = O[k].
$$
The fact that the global observables satisfy the renormalization group equation implies that this map is an isomorphism of cochain complexes, where $\Oo(\E(M))[[\hbar]]$ is equipped with the differential
$$
\what{Q}_k = Q + \{I[k],-\}_1 + \hbar \Delta_k.
$$
The space $\Oo(\E(M))[[\hbar]]$ is naturally a graded commutative algebra. When we also equip $\Oo(\E(M)) [[\hbar]]$ with the Poisson bracket $\{-,-\}_k$ and the differential $\what{Q}_k$, we find that $\Oo(\E(M))[[\hbar]]$ has the structure of a BD algebra.
Using the isomorphism $\op{ev}[k]$, we can transfer this to a BD algebra structure on $\Obs^\ast(M)$. Thus, for each scale $k \in (0,\infty)$, we get a different BD algebra structure on $\Obs^\ast(M)$. These BD algebras are all homotopy equivalent.
The observables thus form a family of BD algebras, but this does not show that they form a BD quantization of the classical observables (see [[the strong quantization theorem]]). We will address this issue in future work.
This wiki will provide the analog, in quantum field theory, of the deformation quantization approach to quantum mechanics. In this introduction, we will start by recalling how deformation quantization works in quantum mechanics.
The collection of observables in quantum mechanics form an associative algebra. The observables of a classical mechanical system form a Poisson algebra. In the deformation quantization approach to quantum mechanics, one starts with a Poisson algebra $A^{cl}$, and attempts to construct an associative algebra $A^q$, which is an algebra flat over the ring $\C[[\hbar]]$, together with an isomorphism of associative algebras $A^q / \hbar \iso A^{cl}$. In addition, if $a,b \in A^{cl}$, and $\til{a}, \til{b}$ are any lifts of $a,b$ to $A^q$, then
$$
\lim_{\hbar \to 0} \frac{1}{\hbar} [\til{a}, \til{b} ] = \{a,b\} \in A^{cl}.
$$
We will describe an analogous approach to studying perturbative quantum field theory. In order to do this, we need to explain the following.
# The structure present on the collection of observables of a //classical// field theory. This structure is the analog, in the world of quantum field theory, of the commutative algebra which appears in classical mechanics. This structure we call a [[commutative factorization algebra]].
# The structure present on the collection of observables of a //quantum// field theory. This structure is that of a [[factorization algebra]]; our definition of factorization algebra is a $\cinfty$ analog of a definition introduced by Beilinson and Drinfeld.
# The extra structure on the commutative factorization algebra associated to a classical field theory which makes it "want" to quantize. This is the analog, in the world of field theory, of the Poisson bracket on the commutative algebra of observables.
# The quantization theorem we prove. This states that, provided certain obstruction groups vanish, the classical factorization algebra associated to a classical field theory admits a quantization. Further, the set of quantizations is pameterized (order by order in $ \hbar $) by the space of deformations of the Lagrangian describing the classical theory.
This quantization theorem is proved using the physicists techniques of perturbative renormalization, as developed in \cite{webbook}. We claim that this theorem is a mathematical encoding of the perturbative methods developed by physicists.
This quantization theorem applies to examples of physical interest, including pure ~Yang-Mills theory and the bosonic string theory. For pure ~Yang-Mills theory, it is shown in \cite{webbook} that the relevant obstruction groups vanish, and that the deformation group is one-dimensional; so that there exists a one-parameter family of quantizations.
Finally, we will explain how (under certain additional hypotheses) the factorization algebra associated to a perturbative quantum field theory encodes the correlation functions of the theory. This justifies the assertion that factorization algebras encode a large part of quantum field theory.
Our goal here is to describe how the observables of a classical field theory naturally form a [[factorization algebra]]. Let $M$ denote a smooth compact manifold (the "spacetime"). We are interested in classical perturbative field theory on $ M $. "Classical" means that the main object of interest is the sheaf of solutions to the ~Euler-Lagrange equations for some local action functional. "Perturbative" means that we will only consider those solutions which are infinitely close to a given solution.
!!! The ~Euler-Lagrange equations
The fundamental objects of a physical theory are the observables of a theory, that is, the measurements one can make in that theory. In a classical field theory, our fields are constrained to be solutions to the ~Euler-Lagrange equations. Thus, the measurements one can make are the functions on the space of solutions to the ~Euler-Lagrange equations.
However, it is essential that we do not take the naive moduli space of solutions. Instead, we consider the //derived// moduli space of solutions. Since we are working perturbatively -- that is, infinitesimally close to a given solution -- this derived moduli space will be a "formal moduli problem" ([[Lur10]]). Thus, the [[first step|elliptic moduli problems]] in our treatment of classical field theory is to develop a language to treat formal moduli problems cut out by systems of partial differential equations on a manifold $M$. Since it is essential that the differential equations we consider are elliptic, we call such an object a //formal elliptic moduli problem//.
Since one can consider the solutions to a differential equation on any open subset $U \subset M$, a formal elliptic moduli problem $\mc{F}$ yields, in particular, a sheaf of formal moduli problems on $M$; which sends $U$ to the formal moduli space $\mc{F}(U)$ of solutions on $U$.
We will use the notation $\EL$ to denote the formal elliptic moduli problem of solutions to the ~Euler-Lagrange equation on $M$; so that $\EL(U)$ will denote the space of solutions on an open subset $U \subset M$.
!!! Observables
In a field theory, we tend to focus on measurements that are localized in spacetime. Hence, we want a method that associates a set of observables to each region in $M$. If $U \subset M$ is an open subset, the observables on $U$ are
$$\Obs^{cl}(U) = \Oo(\EL(U)),$$
our notation for the algebra of functions on the formal moduli space $\EL(U)$ of solutions to the ~Euler-Lagrange equations on $U$. (We will be more precise about which class of functions we are using later). As we are working in the derived world, $\Obs^{cl}(U)$ is a differential-graded commutative algebra. Using these functions, we can answer any question we might ask about the behavior of our system in the region $U$.
The factorization algebra structure arises naturally on the observables in a classical field theory. Let $U$ be an open set in $M$, and $V_1,\ldots,V_k$ a disjoint collection of open subsets of $U$. Then restriction of solutions from $U$ to each $V_i$ induces a natural map
$$ \EL(U) \rightarrow \EL(V_1) \times \cdots \times \EL(V_k).$$
Since functions pullback under maps of spaces, we get a natural map
$$ \Obs^{cl}(V_1) \otimes \cdots \otimes \Obs^{cl}(V_k) \rightarrow \Obs^{cl}(U)$$
so that $\Obs^{cl}$ forms a //pre//factorization algebra. To see that $\Obs^{cl}$ is indeed a factorization algebra, it suffices to observe that the functor $\EL$ is a sheaf.
Since the space $\Obs^{cl}(U)$ of observables on a subset $U \subset M$ is a commutative algebra, and not just a vector space, we see that the observables of a classical field theory form a [[commutative factorization algebra]].
!!! The symplectic structure
Above, we outlined a way to construct, from the elliptic moduli problem associated to the ~Euler-Lagrange equations, a commutative factorization algebra. However, this construction would apply equally well to any system of differential equations. The ~Euler-Lagrange equations, of course, have the special property that they arise as the critical points of a functional.
In finite dimensions, a formal moduli problem which arises as the [[derived critical locus|The classical BV formalism in finite dimensions]] of a function is equipped with an extra structure: a symplectic form of cohomological degree $-1$. For us, this symplectic form is an intrinsic way of indicating that a formal moduli problem arises as the critical locus of a functional. Indeed, any formal moduli problem with such a symplectic form can be expressed (non-uniquely) in this way.
We [[give|The classical BV formalism in infinite dimensions]] a definition of symplectic form on an elliptic moduli problem. We then simply //define// a classical field theory to be a formal elliptic moduli problem equipped with a symplectic form of cohomological degree $-1$.
Given a local action functional satisfying certain non-degeneracy properties, we [[construct|field theories from action functionals]] an elliptic moduli problem describing the corresponding ~Euler-Lagrange equations, and show that this elliptic moduli problem has a symplectic form of degree $-1$.
In ordinary symplectic geometry, the simplest construction of a symplectic manifold is as a cotangent bundle. In our setting, there is a similar construction: given any elliptic moduli problem $\mc{F}$, we [[constrct|cotangent field theories]] a new elliptic moduli problem $T^\ast[-1] \mc{F}$ which has a symplectic form of degree $-1$. It turns out that many examples of field theories of interest in mathematics and physics arise in this way.
!!! The $P_0$ structure
In finite dimensions, if $X$ is a formal moduli problem with a symplectic form of degree $-1$, then the dga $\Oo(X)$ of functions on $X$ is equipped with a Poisson bracket of degree $1$. In other words, $\Oo(X)$ is a [[P_0 algebra|P_0 operad]].
In infinite dimensions, we show that something similar happens. If $\F$ is a classical field theory, then we show that the commutative algebra $\Oo(\F(U)) = \Obs^{cl}(U)$ has a $P_0$ structure; and that the commutative factorization algebra $\Obs^{cl}$ forms a $P_0$ factorization algebra. This is not quite trivial; it is at this point that we need the assumption that our ~Euler-Lagrange equations are elliptic.
Anton Kapustin, //Chiral de Rham complex and the half-twisted sigma-model//, hep-th/0504074
Anton Kapustin and Edward Witten, //Electric-magnetic duality and the geometric Langlands program// arXiv:hep-th/0604151
Our formalism relies heavily on the use of kernels associated to various operators, as is common when working with smooth and distributional sections of vector bundles. Our conventions, however, diverge from the norm because we work with a [[symplectic pairing | symplectic structures on field spaces]] throughout. We give a quick gloss of our conventions and direct the interested reader to chapter 5 of \cite{webbok} for further discussion.
Let $V$ be a $\mathbb Z$-graded vector space with a symplectic pairing $\langle -,-\rangle: V^{\otimes 2} \rightarrow \mathbb R$ of cohomological degree $-1$. Given a kernel $K \in V \otimes V$, we define an operator $K \star: V \rightarrow V$ via the symplectic pairing as follows. The pairing defines a map
\[
1_V \otimes \langle -,-\rangle: V ^{\otimes 3} \rightarrow V,
\]
\[
a \otimes b \otimes c \mapsto a \langle b, c \rangle.
\]
We define
\[
K \star v = (-1)^{|K|} (1_V \otimes \langle -, - \rangle)(K \otimes v).
\]
The notation $|K|$ denotes the cohomological degree of $K$, where we restrict attention here to $K$ of pure degree.
This construction applies to our infinite-dimensional vector spaces as well, of course.
Kontsevich, //Formal (non)-commutative symplectic geometry//, 1993. http://www.ihes.fr/~maxim/TEXTS/Formal%20non-commutative%20symplectic%20geometry.pdf
Let $P$ be an operad in the category of cochain complexes. In this page we will define the notion of //lax $P$-algebra//. (Sometimes people use the term //partial// $P$-algebra instead.)
First, let us recall the definition of a lax symmetric monoidal functor.
''Definition.'' Let $\mc{C}$ be a dg symmetric monoidal category. A lax functor
$$
F : \mc{C} \to \op{Comp}
$$
from $\mc{C}$ to the category of cochain complexes, is a functor $F$, together with product maps
$$
F(a \otimes b) \to F(a) \otimes F(b)
$$
and a unit map
$$
F( 1_{\mc C} ) \to \C
$$
satisfying the following properties.
# The product and unit maps are quasi-isomorphisms.
# The product map is natural in $a$ and $b$.
# These maps satisfy the same symmetry, unit and associativity axioms as are satisfied by an ordinary symmetric monoidal functor.
Let $P$ be any operad. From $P$ we can construct a symmetric monoidal category $\Sym\ P$, described as follows. The objects of $\Sym P$ are $\Z_{\ge 0}$. If $n \in \Z_{\ge 0}$, let $[n] = \{1,2,\ldots, n\}$ denote the corresponding set with $n$ elements. If $n, m \in \Z_{\ge 0}$, then
$$\Sym\ P (n,m) = \oplus_{f : [n] \to [m]} P(f).$$
Here, the direct sum is over surjective maps of finite sets $f : [n] \to [m]$, and
$$
P(f) = \otimes_{i \in [m] } P(f^{-1}(i)).
$$
The composition in $\Sym\ P$ is determined by the operadic composition in $P$.
If $\mc C$ is a symmetric monoidal category, then a $P$-algebra in $\mc C$ is the same thing as a symmetric monoidal functor
$$
\Sym\ P \to \mc C.
$$
''Definition.'' A //lax $P$ algebra// is a lax symmetric functor
$$
\Sym\ P \to \mc C.
$$
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|''License:''|[[BSD open source license|http://tiddlywiki.bidix.info/#%5B%5BBSD%20open%20source%20license%5D%5D ]]|
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In studying field theory, there is a special class of functions on the fields, known as local action functionals, that parametrize the possible physical systems. Let $M$ be a smooth manifold. Let $\E = C^\infty(M,E)$ denote the smooth sections of a $\mathbb{Z}$-graded super vector bundle $E$ on $M$, which has finite rank when all the graded components are included. We call $\E$ the //fields//.
First, we define the space of functions on fields.
''Definition''
//A// functional $F$ //is an element of//
$$
\Oo(\E) = \prod_{n=0}^\infty \operatorname{Hom}_{cont}(\E^{\otimes n}, \mathbb R)^{S_n},
$$
//namely the completed symmetric algebra of $\E^\vee$, the continuous dual space to the field.//
The local functionals depend only on the local behavior of a field, so that at each point of $M$, a local functional should only depend on the jet of the field at that point. Their role is to describe the permitted actions in the Lagrangian formalism for field theory, so we call them //local action functionals//.
''Definition''
//A functional $F$ is// local //if each homogeneous component $F_n$ is a finite sum of terms of the form//
$$
F_n(\phi) = \int_M (D_1 \phi) \cdots D_n(\phi)\, d \mu,
$$
//where each $D_i$ is a differential operator from $\E$ to $C^\infty(M)$ and $d \mu$ is a density on $M$.//
We denote the space of local functionals by $\Ool(\E)$. Often we want to include a formal parameter $\hbar$, so we work with
$$
\Ool (\E)[[\hbar]] := \Ool (\E) \otimes \mathbb R [[\hbar]].
$$
Let $U \subset M$ be an open subset. In this page, we will describe the subspace of the space $\Obs^\ast(M)$ of global strict observables on $M$ which are supported on $U$. The definition of global observables we will use is the one defined using parametrices, and explained [[here|global observables using parametrices]].
We will say that a [[global observable| global observables using parametrices]] is supported on $U$, if the functional $ O[\Phi] $ is supported on $ U $ for $\Phi$ sufficiently small. "Sufficiently small" refers to the partial order on the set of parametrices, where $\Phi < \Phi'$ if $\op{Supp} \Phi \subset \op{Supp} \Phi'$.
Before we make this precise, we need to recall some notation. If $J \in \mathcal{O}(\mathcal{E})[[\hbar]]$, we can expand $J$ as
$$
J = \sum \hbar^i J_{i,k}
$$
where $J_{i,k}$ is a symmetric and continuous multi-linear map
$$
J_{i,k} : \mathcal{E}^{\otimes k} \to \mathbb{R}.
$$
''Definition.''
//A local observable of cohomological degree $d$ is a collection $O[\Phi] \in \mathcal{O}(\mathcal{E})$ of functionals of cohomological degree $d$, for all parametrices $\Phi$, satisfying the following properies.//
* //If $\delta$ is a parameter of square zero and cohomological degree $-d$, then $I[\Phi] + \delta O[\Phi]$ satisfies the renormalization group equation//
$$
I[\Psi] + \delta O[\Psi] = W ( P(\Psi) - P(\Phi) , I[\Phi] + \delta O[\Phi] ).
$$
* //For each $(i,k)$, there exists a compact subset $K \subset U^k$, such that for all $\Phi$ sufficiently small, the $E^{\boxtimes k}$-valued distribution//
$$
O_{i,k}[\Phi] : \mathcal{E}^{\otimes k} \rightarrow \mathbb{R}
$$
//on $M^k$ is supported on $K$.//
//We will let $\Obs^\ast(U)$ denote the graded vector spaces of observables supported on $U$.//
The graded space $\Obs^\ast(U)$ is a sub graded vector space of the cochain complex $\Obs^\ast(M)$. The following lemma implies that this sub graded vector space is preserved by the differential.
''Lemma.''
//Let//
$$
O = \{ O[\Phi] \} \in \Obs^d(U)
$$
//be an observable supported on $ U $. Then, the observable $ \widehat{Q}O \in \Obs^{d+1} (M) $, defined by the collection of functionals//
$$
\left( \widehat{Q} O \right)[\Phi] := Q O[\Phi] + \{ I[\Phi], O[\Phi] \}_{\Phi} + \hbar \Delta_{\Phi} O[\Phi] \in \mathcal{O}(\mathcal{E}) [[\hbar]]
$$
//is supported on $ U $. Thus, $\Obs^{\ast}(U) $ forms a sub cochain complex of $ \Obs^\ast(M) $.//
''Proof.''
The only thing that needs to be checked is the support condition. We need to check that, for each $ (i,k) $, there exists a compact subset $ K $ of $ U^k$ such that, for all sufficiently small $\Phi$, $ \widehat{Q} O_{i,k}[\Phi] $ is supported on $ K$.
Note that we can write
$$
\widehat{Q} O_{i,k}[\Phi] = Q O_{i,k}[\Phi] + \sum_{\substack{a+b = i \\ r + s = k+2 } } \{ I_{a,r}[\Phi], O_{b,s}[\Phi] \}_{\Phi} + \Delta_{\Phi} O_{i-1, k+2}[\Phi].
$$
We know that, for each $ (i,k) $, and for all sufficiently small $\Phi$, $ O_{i,k}[\Phi] $ is supported on $ K$, where $K$ is some compact subset of $ U^k $. It follows that $ Q O_{i,k}[\Phi] $ is suported on $ K $.
By making $ K $ bigger, we can assume that for sufficiently small $\Phi$, $ O_{i-1,k+2}[\Phi] $ is supported on $L$, where $L$ is a compact subset of $U^{k+2}$ whose image in $U^k$, under every projection map, is in $K$. This implies that $ \Delta_{\Phi} O_{i-1, k+2} [\Phi] $ is supported on $ K$.
The locality condition for the effective actions $ I[\Phi] $ implies that, by choosing $\Phi$ to be sufficiently small, we can make $ I_{i,k}[\Phi] $ supported as close as we like to the small diagonal in $M^k$. It follows that, by choosing $\Phi$ to be sufficiently small, the support of $\{ I_{a,r}[\Phi], O_{b,s}[\Phi] \}_{\Phi}$ can be taken to be a compact subset of $U^k$. Since there are only a finite number of terms like $\{ I_{a,r}[\Phi], O_{b,s}[\Phi] \}_{\Phi}$ in the expression for $(\widehat{Q} O)_{i,k}[\Phi]$, we see that, for $\Phi$ sufficiently small, $(\widehat{Q} O)_{i,k}[\Phi]$ is supported on a compact subset of $U^k$, as desired.
$\square$
The local-effective correspondence is one of the central theorems of \cite{webbook}. We first state it for scalar field theories, where the essential idea is easiest to explain, and then we give the most general version, with all its bells and whistles.
!!! Scalar field theories
Recall the notion of a [[quantum field theory]]. We will work here with a smooth compact Riemannian manifold $M$ and scalar fields on $M$. We have a bijection between the set of scalar field theories and the set of [[ local action functionals|local action functional]].
''Theorem''
//Let $\mathcal{T}^{(n)}$ denote the set of perturbative quantum field theories defined modulo $\hbar^{n+1}$. Then $\mathcal{T}^{(n+1)}$ is, in a canonical way, a principal bundle over $\mathcal{T}^{(n)}$ for the abelian group $\mathcal{O}_{loc}(C^\infty(M))$ of local action functionals on $M$. Further, $\mathcal{T}^{(0)}$ is canonically isomorphic to the space $\mathcal{O}^+_{loc}(C^\infty(M))$ of local action functionals which are at least cubic.//
The theorem tells us that we can convert any interaction term $I$ inside an action of the form
$$
S(\phi) = \int_M \phi (D + m^2) \phi + I(\phi),
$$
into an effective field theory, and vice versa. More precisely, this theorem says that the two spaces are equivalent, but to specify a bijection, one must pick a //renormalization scheme//, which allows one to use counterterms to turn a local action functional into an effective field theory.
!!! The general case
Here we allow ourselves to work on a noncompact smooth Riemannian manifold $M$ and we consider families of theories parametrized by a nilpotent graded manifold $(X,A)$ (the nilpotent directions are useful for studying deformations of theories). We use the notation from [[quantum field theory]]. Thus, on $M$ we have a presheaf of theories $\mathcal{T}^{(n)}(\mathcal{E},\mathcal{A})$ defined modulo $\hbar^{n+1}$.
''Theorem''
# //The presheaves $\mathcal{T}^{(n)}(\mathcal{E},\mathcal{A})$ and $\mathcal{T}^{(\infty)}(\mathcal{E},\mathcal{A})$ are in fact sheaves.//
# //The sheaf $\mathcal{T}^{(0)}(\mathcal{E},\mathcal{A})$ is canonically isomorphic to the subsheaf of local functionals $\mathcal{O}_{loc}(\mathcal{E}_c, \mathcal{A})$ which are at least cubic modulo the ideal $\Gamma(X,\mathfrak{m}) \subset \mathcal{A}$.//
# //For $n > 0$, the sheaf $\mathcal{T}^{(n)}(\mathcal{E},\mathcal{A})$ is a torsor over $\mathcal{T}^{(n-1)}(\mathcal{E},\mathcal{A})$ for the sheaf of abelian groups $\mathcal{O}_{loc}(\mathcal{E}_c, \mathcal{A})$ in a canonical way.//
This theorem tells us several wonderful things. First, we can construct field theories in a local-to-global manner. Second, we can convert bijectively between local action functionals and theories.
There is a special class of factorization algebras that are more tractable in the same way that locally constant sheaves are easier to work with than arbitrary sheaves.
''Definition''
A factorization algebra $\F$ on an $n$-manifold $M$ is //locally constant// if for every inclusion of an open ball $D \cong \R^n$ into a larger ball $D'$, the map $\F(D) \to \F(D')$ is a quasi-isomorphism of cochain complexes.
Many examples arise from topology, such as labelled configuration spaces (as discussed in the work of Segal, ~McDuff, Bodigheimer, Salvatore, and Lurie).
''Proposition''
An $E_n$ algebra $A$ defines a locally constant factorization algebra $\F_A$ on $\R^n$, where $\F_A(D) = A$ on a ball $D \subset \R^n$ and the $E_n$ structure provides the structure maps of $\F_A$ for inclusion of tuples of balls into a larger ball.
As the open balls in $\R^n$ provide a factorizing basis for $\R^n$, we naturally extend $\F_A$ to a locally constant factorization algebra.
Lurie, Derived algebraic geometry V: structured spaces
Lurie, Moduli problems for ring spectra, 2010
<<getTiddlerPassword>>
The model problems of classical and quantum mechanics involve a particle moving in some Euclidean space $\mathbb{R}^n$ under the influence of some fixed field. Our main goal in this page is to describe these model problems in a way that makes the idea of a [[factorization algebra]] emerge naturally, but we also hope to give mathematicians some feeling for the physical meaning of terms like "field" and "observable." We will not worry about making precise definitions, since that's what this site aims to do. As a narrative strategy, we describe a kind of cartoon of a physical experiment, and ask that physicists accept this cartoon as a friendly caricature elucidating the features of physics we most want to emphasize.
!! A particle in a box
!!!! A first pass
Suppose we want to study the physical behavior of some type of particle. We confine our particle inside some box and occasionally take measurements of this system. The set of possible trajectories of the particle around the box constitute all the imaginable behaviors of this particle; we might write this mathematically as $Maps(I,box)$, where $I \subset \mathbb{R}$ denotes the time interval over which we conduct the experiment. In other words, the set of possible behaviors forms a space of //fields// on the timeline of the particle.
The behaviour of our theory is governed by the action functional. The simplest case is the action of the free field theory, whose value on a function $f: I \to box$ is
$$
S(f) = \int_{I} \ip{ f , \Delta f }.
$$
!!!!! Measurements
The notion of measurement is fraught in physics, but we will take a very concrete view. Taking a measurement means that we have physical measurement device (e.g., a camera) that we allow to interact with our system for a period of time. The measurement is then how our measurement device has changed due to the interaction. In other words, we //couple// the two physical systems, then decouple them and record how the measurement device has modified from its initial condition. (Of course, there is a symmetry in this situation: both systems are affected by their interaction, so a measurement inherently disturbs the system under study.)
The //observables// for a physical system are all the imaginable measurements we could take of the system. Instead of considering all possible observables, we might also consider those observables which occur within a specified time period.
!!!!! Classical field theory
Let us start by considering the classical case. Then, the trajectory of the particle is constrained to be in a solution to the ~Euler-Lagrange equations of our theory. For example, if the action functional governing our theory is the action of the free theory, then a map $f : I \to box$ satisfies the ~Euler-Lagrange equation if it is a straight line.
We are interested in the observables for this classical field theory. Since the trajectory of our particle is constrained to be a solution to the ~Euler-Lagrange equation, the only measurements one can make are functions on the space of solutions to the ~Euler-Lagrange equation.
If $U \subset \mathbb{R}$ is an open subset, we will let $ \operatorname{Fields (U)} $ denote the space of fields on $ U $, that is, the space of maps $ f : U \rightarrow box $. We will let
$$
\operatorname{EL}(U) \subset \operatorname{Fields }(U)
$$ denote the subspace consisting of those maps $f : U \to box$ which are solutions to the ~Euler-Lagrange equation. As $U$ varies, $\operatorname{EL}(U)$ forms a sheaf of spaces on $\mathbb{R}$.
We will let $\operatorname{Obs}^{cl}(U)$ denote the space of functions on $\operatorname{EL}(U)$ (the precise class of functions we will consider will be discussed later). As $U$ varies, the spaces $\operatorname{Obs}^{cl}(U)$ form a cosheaf of commutative algebras on $\mathbb{R}$. We will think of $ \operatorname{Obs}^{cl}(U) $ as the space observables for our classical system which only consider the behaviour of the particle on times contained in $ U $.
!!!!! Quantum field theory
Measurements in quantum field theory are, of course, much more subtle. The most naive guess is to postulate that the space of observables one assigns to an open subset $ U\subset \mathbb{R} $ is the space $\mathcal{O}( \operatorname{Fields}(U)) $ of functions on the space of fields on $ U $. Thus, we will set
$$
\operatorname{Obs }^{naive}(U) = \mathcal{O} ( \operatorname{Fields }(U) ).
$$
A moments thought tells us that this can not be the correct answer. After all, in the classical setting, the space of observables is far smaller: it is a space of functions on the subspace $ \mathcal{EL}(U) \subset \operatorname{Fields }(U) $. By the correspondence principle, one would expect that the space of observables for the quantum theory on $ U $ will reproduce the space of observables for the classical theory in the $ \hbar \rightarrow 0 $ limit.
Let us assume, however, that $ \operatorname{Obs }^{naive}(U) $ maps onto the space of quantum observables: that is, we will assume that every reasonable function on the space $ \operatorname{Fields }(U) $ will define a measurement of our system.
If $ U_1 $
Then, by taking the expectation value over many m
The //observables// for a physical system are all the imaginable measurements we could take of the system. As a mathematical idealization, we might write $Obs = Maps(Fields, \mathbb{R})$. But we might also want to understand "when" certain observations occur: we might like to know the position of the particle at some moment in time. More formally, if we pick a subinterval of time $J \subset I$, we might consider observables that only depend on the behavior of the particle during $J$. This space of observables $Obs(J) = Maps(Maps(J,box), \mathbb{R})$ is a subspace of all the observables $Obs(I) = Maps(Maps(I,box), \mathbb{R})$ because of how functions pull back:
$$
J \overset{i}{\hookrightarrow} I \Longrightarrow Maps(I,box) \overset{i^\ast}{\rightarrow} Maps(J,box) \Longrightarrow Obs(J) \overset{(i^\ast)^\ast}{\rightarrow} Obs(I).
$$
Thus the observables behave like a cosheaf on the timeline of the particle: if one interval is a subset of a bigger interval, the observables on the smaller interval yield observables on the bigger interval. Moreover, we can combine observables from disjoint intervals. Let $J$ and $J'$ be disjoint subintervals of $I$. Then $Maps(J \cup J',box) = Maps(J,box) \times Maps(J',box)$ and so
$$
Obs(J \cup J') = Maps(Maps(J,box) \times Maps(J',box), \mathbb{R}) = Obs(J) \otimes Obs(J')
$$
This relationship means that if we pick an observable $F$ on $J$ and an observable $F'$ on $J'$, then $F \otimes F'$ yields an observable on $J \cup J'$.
Measurements (and so observables) differ qualitatively in the classical and quantum settings. If we study a classical particle, the system is not noticeably disturbed by measurements, and so we can do multiple measurements at the same time. Hence, on each interval $J$ we have a commutative multiplication map $Obs(J) \otimes Obs(J) \rightarrow Obs(J)$, as well as the maps that let us combine observables on disjoint intervals. For a quantum particle, however, a measurement disturbs the system significantly, so that
# it matters in which order we make our measurements (e.g., the famous relation "$pq - qp \neq 0$");
# taking two measurements simultaneously is incoherent, as the measurement devices are coupled to each other and thus also affect each other, so that we are no longer measuring just the particle.
Quantum observables thus do not form a cosheaf of commutative algebras on the interval; we only have the maps that let us combine observables on disjoint intervals. Moreover, in the quantum setting, we only expect to see clear patterns in the //average// behavior of the particle; in contrast, the classical particle behaves in a completely structured way once make enough observations. For quantum physics, we thus need to know how to take the //expectation value// of any observable.
!!!! Taking the action functional into account
Our description of the situation above ignores all dynamical aspects of physics. We explained the space of possible behaviors and possible measurements -- what we might call the kinematics -- but we did not describe the rules or equations that govern the actual behavior and measurements -- so to speak, the dynamics. This ingredient comes in the form of an action functional, ~Euler-Lagrange equations, or a Hamiltonian.
In the experimental setting, we might know nothing //a priori// about the dynamics. The goal of the experiments is to discover patterns in the particle's behavior. For a classical system, we might run our experiment many times, modifying the initial conditions each time. Given enough data, we would discover that the behavior of our particle is described by some subspace of the space of all possible trajectories: each initial condition leads to a unique trajectory, and we can parametrize the actual behavior by these initial conditions using -- typically -- some system of differential equations.
In the theoretical setting, we generally start with some functional $S$ on the space of fields, known as the //action functional//, and the critical points of $S$ constitute the subspace of actual behavior inside the the full space of fields. This action functional arises usually from some combination of physical reasoning (e.g., symmetry arguments) and experimental data.
Since the space of actual behavior is a subspace of the all possible behaviors, the algebra of functions on this subspace is a quotient of the algebra of functions on all fields. Typically, this subspace corresponds to the solution set of the ~Euler-Lagrange equations arising from our action functional, and so these ~Euler-Lagrange equations determine the ideal whose zero set is the subspace of actual trajectories. Two types of measurements that alway yield the same answer for our physical system ought to be identified. For example, in a system where energy is conserved, if we measure the energy of the particle at two different times, we get the same answer. Thus the observable "energy at time $t$" and the observable "energy at time $T$" will always give the same answer, and so they are equivalent observables for our system. The quotient of $Obs$ from above by this relation is thus the natural algebra of observables for our classical system.
In summary, classical physics fits nicely into the framework of geometry as the observables form a cosheaf of commutative algebras.
The quantum situation is more subtle, since it doesn't fit into the usual framework of geometry (factorization algebras aim to provide the necessary framework, but we're getting ahead of ourselves). Again, after doing many experiments, we hope that patterns will emerge in the average values of the observables. The correspondence principle asserts that these average values ought to mimic the behavior of the analogous classical system.
I'M STUCK!
!! Putting it into mathematics
We will now make some statements from above more precise to foreshadow the constructions made on the rest of the site. Keeping this example in mind will motivate nearly all our work.
Again, we start with the kinematics. We idealize the box as some Euclidean space $\mathbb{R}^n$ and we restrict attention to //smooth// trajectories, so our fields are $Fields = \mathcal{E} = C^\infty(I,\mathbb{R}^n)$ where $I$ is some interval in $\mathbb{R}$. Notice that we can view the fields as forming a sheaf of vector spaces on $I$: to each open subinterval $J \subset I$, we assign $\mathcal{E}(J) = C^\infty(J,\mathbb{R}^n) \cong C^\infty(J)^{\oplus n}$.
We now want to describe the observables for this space of fields. We'll give the mathematical description and then give some physical justification. From a mathematical viewpoint, the observables are just the functions on the space of fields. For our situation, the fields form a vector space, and the simplest class of functions on a vector space $V$ are the polynomials, namely $\mathcal{O}(V) = Sym (V^\vee)$, the symmetric algebra of the dual space $V^\vee$. When the vector space is the smooth functions $C^\infty(J)$, the natural dual space is the compactly supported distributions $D(J)$, which are the continuous linear functionals on smooth functions. Notice that the linear functionals on the sheaf of fields thus forms a cosheaf, namely the cosheaf of compactly-supported distributions $\mathcal{E}^\vee(J) \cong D(J)^{\oplus n}$ dual to our smooth functions. Hence, the observables assign $Sym (\mathcal{E}^\vee(J))$ to each interval $J$.
Notice that these observables behave in accordance to our heuristic description above. Given an inclusion of intervals $J \overset{i}{\hookrightarrow} J'$, then smooth functions pull back and hence compactly supported distributions push forward. The map $D(J) \hookrightarrow D(J')$ induces a map $Obs(J) = Sym (D(J)) \rightarrow Sym (D(J')) = Obs(J')$. Moreover, given two disjoint open intervals $J$ and $J'$, we know $D(J \cup J') = D(J) \oplus D(J')$ and hence we see
$$
Obs(J \cup J') = Sym(D(J) \oplus D(J')) = Sym(D(J)) \otimes Sym(D(J')) = Obs(J) \otimes Obs(J').
$$
Thus we have canonical maps that allow us to combine observables with disjoint support.
//Remark:// Throughout, we use the homomorphisms and tensor products appropriate to smooth functions and distributions. That is, we always use continuous linear maps and the completed projective tensor product.
At the very least these observables are a reasonable approximation to the observables that physicists want to use. For instance, the basic measurements physicists might want to make are "the position at time $t$" or "the square of the average acceleration over the interval $J$." Mathematically, we can express these as
$$
\phi \mapsto \phi(t_0) = \delta_{t_0}(\phi)
$$
and
$$
\phi \mapsto \left( \int_J \ddot{\phi}(t) dt \right)^2
$$
where $\phi$ is a field. Notice that both these operators are polynomials of distributions and so live in $Sym(D(I))$.
Now we will include the dynamics. Given a system of ~Euler-Lagrange equations, they impose relations among elements of $Sym(D(U))$. For example, if we have a fixed force field $F$, then we typically have Newton's equation $\ddot(phi)(t_0) = F(t_0)$ for every point $t_0 \in I$. For this system, the observables at the point $t_0$ are just $\mathbb{R}[\delta_{t_0},\dot{\delta}_{t_0}]$, the polynomial ring generated by the position and velocity operators.
STUCK AGAIN
!! Where we go from here
We have seen that the observables of a field theory have an interesting structure as we look at different open sets in our spacetime manifold $M$ (e.g., the timeline of our particle). Namely, let $Obs$ denote the functor that assigns a vector space $Obs(U)$ to each open set $U \subset M$. Then we noticed the following basic properties
# an inclusion of open sets $U \hookrightarrow V$ yields a map of observables $Obs(U) \rightarrow Obs(V)$;
# the observables on a disjoint pair of open sets $Obs(U \cup V$ is isomorphic to the tensor product $Obs(U) \otimes Obs(V)$.
We axiomatize these properties using the notions of [[prefactorization algebra]] and [[factorization algebra]], and part of this site is devoted to exploring the resulting mathematical strucutures. But our primary aim is to use this mathematical language to describe the observables of a quantum field theory. In particular, this language allows us to treat quantum field theory in the style of deformation quantization.
In this section, we will give a definition of the commutative factorization algebra of observables of a [[classical field theory|definition of classical field theory]].
If $U \subset M$ is an open subset, let
$$
\Obs^{cl}(U) = \Oo( \E(U))
$$
be the graded algebra of functions on the space $\E(U)$ of fields on $U$.
[[Recall|symplectic structures on field spaces]] that we can define the Poisson bracket of a local functional $S \in \Ool(\E_c(U))$ with any functional $\Phi \in \Oo(\E(U))$. Define a differential on $\Obs^{cl}(U)$ by
$$\begin{split}
\d : \Oo(\E(U)) = \Obs^{cl}(U) & \to \Obs^{cl}(U) \\
\d \Phi &= \{S,\Phi\}.
\end{split}
$$
The fact that $S$ is of cohomological degree $0$ and satisfies the classical master equation implies that $\d$ is of cohomological degree $1$ and square zero.
Note that $\d$ is a derivation for the natural commutative algebra structure on $\Oo(\E(U))$. If $U \subset V$, there is a natural inclusion map
$$
\Oo(\E(U)) \to \Oo(\E(V))
$$
dual to the restriction map on sections $\E(V) \to \E(U)$.
''Lemma''
The map
$$
\Oo(\E(U)) \to \Oo(\E(V))
$$
is a map of commutative differential graded algebras.
Thus, sending $U \to \Obs^{cl}(U)$ defines a pre-cosheaf of differential graded commutative algebras, and so a commutative prefactorization algebra.
''Theorem''
The prefactorization algebra $\Obs^{cl}$ is a factorization algebra.
[[Proof|Proof that classical observables are a factorization algebra]].
Throughout this site we take the path integral formalism as fundamental, and hence we do not focus on the Hamiltonian, or operator, approach to quantum physics. On this page, however, we will explain how to express the standard formalism of quantum mechanics in the language of factorization algebras.
!!! Quantum mechanics
//Remark:// Our goal here is to sketch the formal aspects of quantum mechanics, so we avoid technical issues (such as boundedness of operators or whether they are trace-class) by describing the simpler finite-dimensional setting.
Let $V$ denote a finite-dimensional Hilbert space, $A$ the continuous endomorphisms, and $H \in A$ the Hamiltonian operator. We view $V$ as a state space for our system, $A$ as where the observables live, and $H$ as determining the time evolution of our system. We seek to describe the following experimental situation, which one might view as a scattering experiment:
* at time $t = 0$, we prepare our system in the initial state $ v_0 \in V$;
* we modify the governing Hamiltonian over some finite time interval (i.e., apply an operator aka observable);
* at time $t = T$, we measure whether our system is in the final state $v_ 1 \in V$.
If we run this experiment many times, with the same initial and final states and the same operator, we should find a statistical pattern in our data. To put this in the usual Dirac notation, if we denote the operator by $\mathcal{O}$ and idealize it as happening at a fixed moment $t_0$ in time, then we are trying to compute the number
$$
\langle v_1 | e^{i(T-t_0)H} \mathcal{O} e^{it_0 H}|v_0\rangle.
$$
//Remark:// A state is actually a ray, or line, in $V$. We address this issue in the next section.
We want to describe this situation using a factorization algebra $F$ on the interval $[0,T]$. Before jumping into details, here's the guiding idea. Interior open intervals describe moments when operators can act on our system. An interval that contains $0$ (but not the other end) should describe a state of the system (and dually for intervals containing the other endpoint). But not only do we assign $V$ to a connected interval containing exactly one endpoint and $A$ to a connected interior interval; we also have a distinguished vector in each of these vector spaces. It is important to recall that a factorization algebra $F$ always assigns a "pointed" vector space to an open set: $F(\emptyset) = \mathbb{C}$ and so the inclusion of the empty set into an open $U$ always gives a map $\mathbb{C} \rightarrow F(U)$. Since the empty set has empty intersection with itself, we see that we have a distinguished map $F(\emptyset) \otimes F(\emptyset) \rightarrow F(\emptyset)$, namely multiplication. Hence we assign $\mathbb{C}$ //with multiplication// to the empty set, and so we have a distinguished element in $F(\emptyset)$, namely $1$, which picks out a distinguished vector in every $F(U)$. Thus, a factorization algebra assigns a pair $(F(U), u \in F(U))$ to each open set.
Returning to quantum mechanics, we fix a vector $v_0 \in L_0$ and a vector $v_1 \in L_1$. To open subintervals, our functor assigns the following vector spaces:
* $[0,t) \mapsto (V, e^{itH} v_0)$
* $(s,t) \mapsto (A, e^{i(t-s)H})$
* $(t,T] \mapsto (V, e^{-i(T-t)H}v_1)$.
(We defer describing $F([0,T]$.) The first type of interval describes how the initial condition has evolved up to time $t$; the interior interval describes the possible operators that can act during that time interval, and the evolution operator is the natural distinguished operator; the final type of interval describes the state that will evolve to the final condition.
We must now describe the maps coming from inclusion of intervals. Hopefully, we give enough examples to pin down the idea.
First we describe inclusions of connected intervals.
* For $[0,s) \subset [0,t)$, we use the map $v \mapsto e^{i(t-s)H}v$. This is an automorphism of $V$ sending $e^{isH}v_0$ to $e^{itH}v_0$, and so respects our marked points.
* For $(s,t) \subset (s',t')$, we use $\mathcal{O} \mapsto e^{i(t'-t)H} \mathcal{O} e^{i(s-s')H)}$.
* For $(s,t) \subset [0,t')$, we use $\mathcal{O} \mapsto e^{i(t'-t)H} \mathcal{O} e^{isH}v_0$. This sends an operator in $A$ to how it acts on the input state $v_0$.
* For $(s,t) \subset (s',T]$, we use $\mathcal{O} \mapsto e^{-i(s-s')H} \mathcal{O}^\dagger e^{-i(T-t)H}v_1$.
Next we describe the three simplest types of "multiplication," namely a pair of disjoint intervals maps into a bigger, connected interval.
* For $[0,s) \cup (s',t) \subset [0,t)$, we use $v \otimes \mathcal{O} \mapsto \mathcal{O} e^{i(s'-s)H}v$. This map describes how an operator acts on a state.
* For $(s,s') \cup (t,t') \subset (s,t')$, we use $\mathcal{O} \otimes \mathcal{P} \mapsto \mathcal{P} e^{i(t-s')H} \mathcal{O}$.
* For $(s,s') \cup (t,T] \subset (s,T]$, we use $\mathcal{O} \otimes v \mapsto \mathcal{O}^\dagger e^{-i(t-s')H}v$.
Finally, we describe what our functor assigns to the whole interval $[0,T]$. Using the gluing axiom, we see that it assigns the vector space $V \otimes_A V$, where the lefthand $V$ corresponds to the incoming states (and $A$ acts by multiplication) and the righthand $V$ corresponds to the outgoing state (and $A$ acts by multiplication of the adjoint). Hence we can view this as $V \otimes_{End(V)} V^\vee$, which is isomorphic to the ground field $\mathbb{C}$: we have the map
$$
V \otimes_{\mathbb{C}} A \otimes_{\mathbb{C}} V \rightarrow V \otimes_A V
$$
$$
v \otimes \mathcal{O} \otimes w \mapsto \langle w | \mathcal{O} v \rangle,
$$
the usual inner product!
The maps in this factorization algebra thus allow us to compute scattering as follows. Suppose we apply some operator during the interval $(s,t)$. The inclusion of intervals $(s,t) \hookrightarrow [0,T]$ yields a map $A \rightarrow V \otimes_A V \cong \mathbb{C}$. If we pick an operator $\mathcal{O}$ acting during the interval $(s,t)$, then its image in $\mathbb{C}$ is $\langle v_1 | e^{i(T-t)H} \mathcal{O} e^{isH} | v_0 \rangle$.
!!! Some subtleties
Our construction above captures much of the standard formalism of quantum mechanics, but there are a few loose ends we need to address.
First, in standard quantum mechanics, a state is not a vector in $V$ but a line. Above, however, we fixed vectors $v_0$ and $v_1$, so there seems to be a discrepancy. The observation that rescues us is a natural one, from the mathematical viewpoint. Consider scaling $v_0$ and $v_1$ by elements of $\mathbb{C}^\times$. This defines a new factorization algebra, but it is isomorphic to what we described above, and the expectation value "$\langle v_0 | \mathcal{O} v_1 \rangle$" of an operator depends linearly in the rescaling of the input and output vectors. More precisely, there is a natural equivalence relation we can place on the factorization algebras described above that corresponds to the usual notion of state in quantum mechanics.
Another issue that might bother the reader is that our formalism only matches nicely with experiments that resemble scattering experiments. It does not seem well-suited to descriptions of systems like bound states (e.g., an atom sitting quietly, minding its own business). For such systems, we might consider running over the whole space of states (as described in the previous paragraph). Alternatively, we might drop the endpoints and simply work with the factorization algebra on the open interval, which focuses on the algebra of operators $A$.
This paper provides the analog, in quantum field theory, of the deformation quantization approach to quantum mechanics. So far we have discussed how the notion of factorization algebra provides a language for observables of a field theory. We now want analogs for factorization algebras of being a Poisson algebra or a deformation of an algebra in order to formulate a deformation quantization theorem.
The basic philosophy is the following:
# The observables of a classical field theory form a $P_0$ factorization algebra, $\Obs^{cl}$. (A $P_0$ factorization algebra is a kind of Poisson structure as discussed in the [[P_0 operad]].)
# A quantization of a classical field theory is given by a factorization algebra $\Obs^q$, flat over $\R[[\hbar]]$, such that $\Obs^q(U)$ is a quantization of the $P_0$ algebra of classical observables on $U$. Thus, we require there is a quasi-isomorphism of factorization algebras $\Obs^q \otimes_{\R[[\hbar]]} \R \simeq \Obs^{cl}$ such that the bracket on $\Obs^{cl}$ defined by $\Obs^q$ coincides, up to a specified homotopy, with that given by the natural $P_0$ structure on $\Obs^{cl}$.
On this wiki we do the following:
# From a classical field theory on a manifold $M$ -- given, in the usual way, by a Lagrangian, and possibly with symmetries such as gauge symmetries -- we construct a $P_0$ factorization algebra of observables of the theory.
# From a quantum field theory in the sense of \cite{webbook}, we construct yields a BD factorization algebra of quantum observables. This BD factorization algebra is a quantization of the $P_0$ factorization algebra constructed from the corresponding classical theory.
The techniques of \cite{webbook} provide an obstruction-theoretic framework for constructing quantum field theories. Thus, when coupled with the results of this paper, we have a mechanism to construct factorization algebras from classical field theories, using obstruction theory.
[[Later | refined quantization theorem]] we will introduce a stronger definition of quantization, where the quantized factorization algebra has an additional operadic structure. We conjecture that, with this stronger definition, all quantizations arise in this way.
Further, we believe that the factorization algebra associated to a quantum field theory encodes most of the structure of the theory. For instance, from the factorization algebra, one can recover things like the operator product expansion (OPE) of observables. In good cases, one can also recover the [[correlation functions]] of the theory.
Throughout this wiki, we work in the symmetric monoidal category of complete nuclear spaces. Although many natural spaces in analysis are nuclear (e.g., smooth functions on a compact manifold), the lovely formal properties of nuclear spaces may be unfamiliar, so we provide here a quick guide.
The two essential facts to bear in mind are that
# the natural function spaces for geometry (rather than hard analysis) are usually nuclear, such as smooth functions, distributions, and Schwartz functions or distributions;
# nuclear spaces behave much like finite-dimensional vector spaces, particularly with respect to $Hom$ and $\otimes$.
We briefly review the major definitions, state the important properties, describe the relevant examples, and then explain why nuclear Frechet spaces provide the appropriate context for this work.
!! Basic definitions
Recall that a //topological vector space// (sometimes abbreviated to TVS) over $\mathbb{R}$ or $\mathbb{C}$ is a vector space equipped with a topology that makes scalar multiplication and addition continuous. The most important class of TVS (containing both Banach and nuclear spaces) are the //locally convex spaces//. Such a space has a basis (for the topology) given by convex sets. Finally, a Frechet space is a complete, metrizable locally convex spaces.
The subtleties in functional analysis often arise from the freedom to choose different topologies on dual spaces or tensor products of spaces. For $V$ a TVS, let $V^\ast$ denote the space of //all// linear maps from $V$ to the base field, and let $V^\vee \subset V^\ast$ denote the subspace of //continuous// linear functionals. There are several natural topologies on the $V^\vee$, of which two are the following. The //weak// topology on $V^\vee$ is the topology of pointwise convergence: a sequence of functionals $\{\lambda_n\}$ converges to zero if the sequence $\{\lambda_n(x)$ converges to zero for any $x \in V$. The //strong// topology on $V^\vee$ is the topology of bounded convergence: a [*http://en.wikipedia.org/wiki/Filter_(mathematics) filter] of the origin converges in the strong topology if it converges uniformly on every bounded subset in $V$. We will use the strong topology here.
There are also several natural topologies one can put on the space $\op{Hom}(E,F)$ of continuous linear maps from $E \to F$. The topology of interest here is the topology of bounded convergence, described as follows. Let $U \subset F$ be an open subset of $F$, and $ B \subset E$ a bounded subset of $E$. A basis of neighbourhoods of $0$ in $\Hom(E,F)$ consists of those subsets
$$\mc{U}(B,U) = \{f : E \to F \mid f(B) \subset U \} \subset \Hom(E,F).$$
With this topology, as long as both $E$ and $F$ are locally convex and Hausdorff, so is $\Hom(E,F)$.
We define the projective topological tensor product of two locally convex spaces $V$ and $W$ as follows. Let $V \otimes_{alg} W$ denote the algebraic tensor product of $V$ and $W$. The //projective// topology on $V \otimes_{alg} W$ is the finest locally convex topology so that the canonical map
$$
V \times W \to V \otimes_{alg} W
$$
is continuous.
We will let $V \otimes_\pi W$ (or just $V \otimes W$ when there is no ambiguity) denote the completion of $V \otimes_{alg} W$ with respect to this tensor product.
There is another natural topology on $V \otimes_{alg} W$, called injective topology, which is coarser than the projective topology; every reasonable topology on $V \otimes_{alg} W$ lies between these two. (We will not recall the precise definition of the injective topology, as it is a little technical: see [Treves] or [Grothendieck] for details).
We will let $V \otimes_i W$ denote the completion of $V \otimes_{alg} W$ with respect to the injective topology.
We now give the most elegant definition of a nuclear space:
''Definition''
//A locally convex Hausdorff space $V$ is// nuclear //if the canonical map $V \hat{\otimes}_p W \rightarrow V \hat{\otimes}_i W$ is an isomorphism for any locally convex Hausdorff space $W$.//
There are several equivalent definitions, although these equivalences are highly nontrivial. For more, see [*http://en.wikipedia.org/wiki/Nuclear_space nuclear space] or chapter 50, \cite{treves}.
Throughout this wiki, we will work in the following category, whose nice properties are described below.
''Definition''
//Let $Nuc$ denote the category, enriched in ordinary vector spaces, whose objects are complete nuclear spaces and whose morphisms are linear continuous maps.//
//Remark:// Although the space $\Hom(E,F)$ of continuous linear maps between nuclear spaces does have a natural topology, as described above, $Nuc$ does not form a symmetric monoidal category enriched in topological vector spaces.
If $E,F$ are nuclear spaces, then so is their completed tensor product $E \otimes F$. Thus, we have
''Proposition''
$Nuc$ //is a symmetric monoidal category with respect to the completed tensor product.//
!! Examples
Many function spaces in geometry are nuclear:
* smooth functions on an open set in $\mathbb{R}^n$,
* smooth functions with compact support on an open set, or on a compact set, in $\mathbb{R}^n$,
* distributions (the continuous dual space to compactly-supported smooth functions) on an open set in $\mathbb{R}^n$,
* distributions with compact support on an open set in $\mathbb{R}^n$,
* Schwartz functions on $\mathbb{R}^n$,
* Schwartz distributions on $\mathbb{R}^n$,
* holomorphic functions on an open set in $\mathbb{C}^n$,
* holomorphic distributions on an open set in $\mathbb{C}^n$,
* formal power series in $n$ variables (with the inverse limit topology),
* polynomials in $n$ variables.
In consequence, we see that, for example, smooth sections of a vector bundle on a compact manifold form a nuclear Frechet space.
!! Important properties
Nuclear spaces enjoy the following useful properties:
* A linear subspace of a nuclear space is nuclear.
* The quotient of a nuclear space by a closed linear subspace is nuclear.
* A countable direct sum of nuclear spaces (equipped with finest locally convex topology) is nuclear.
* Likewise, a countable colimit of nuclear spaces is nuclear.
* A direct product of nuclear spaces (equipped with the product topology) is nuclear.
* Likewise, a limit of nuclear spaces is nuclear.
* The completed tensor product is nuclear.
* If $E = \lim E_i$, $F = \lim F_j$ are nuclear spaces written as limits of nuclear spaces $E_i, F_j$, then
$$
E \otimes F = \lim_{i,j} (E_i \otimes F_j).
$$
!! Subcategories
Let us list three other important classes of nuclear spaces:
# ($\mc{NF}$) spaces, that is nuclear Frechet spaces.
# ($\mc{NL F}$) spaces, which are countable colimits of $\mc{NF}$-spaces.
# ($\mc{NDF}$) spaces, that is, nuclear ($\mc{DF}$) spaces. A space is ($\mc{NDF}$) if and only if its strong dual is of type ($\mc{NF}$).
* If $E$ is a nuclear Frechet space (that is, of type ($\mc{NF}$)), then the strong dual $E^\vee$ is a space of type ($\mc{NDF}$).
* If $E$ is of type ($\mc{NDF}$), then the strong dual $E^\vee$ is of type ($\mc{NF}$).
* Countable products of spaces of type ($\mc{NF}$) are again of type ($\mc{NF}$).
* Closed subspaces and quotients of spaces of type ($\mc{NF}$) are again of type ($\mc{NF}$) (thus, spaces of type ($\mc{NF}$) are closed under countable limits and finite colimits).
* If $E,F$ are both of type ($\mc{NF}$), or both of type ($\mc{DF})$, then
$$
(E \otimes F)^\vee = E^\vee \otimes F^\vee.
$$
* If $E,F$ are both of type ($\mc{NF}$) (or of type ($\mc{NDF}$)) then so is $E \otimes F$.
* If $E = \colim E_i$ is a countable colimit of nuclear spaces, and $F$ is of type ($\mc{NDF}$), then
$$
E \otimes F = \colim (E_i \otimes F).
$$
* If $M$ is a manifold, and $F$ is any locally convex Hausdorff topological vector space, then $\cinfty(M) \otimes F$ is the space of smooth maps $M \to F$.
* If $M$ and $N$ are manifolds, then
$$
\cinfty(M) \otimes \cinfty(N) = \cinfty( M \times N).
$$
* Any nuclear space is semi-reflexive: this means that the natural map $E \to \left(E^\vee\right)^\vee$ is a linear isomorphism (although not necessarily a homeomorphism).
* Any nuclear space $E$ of type ($\mc{NF}$) is reflexive: this means that the natural map $E \to \left(E^\vee\right)^\vee$ is a linear homeomorphism, where duals are given the strong topology. This follows from the facts that ($\mc{NF}$) spaces are barrelled (Treves, page 347); barrelled nuclear spaces are Montel (Treves, page 520); and Montel spaces are reflexive (Treves, page 376).
* If $E$ is a nuclear space of type ($\mc{NF}$), then the strong and weak topologies on bounded subsets of $E^\vee$ coincide (Treves, page 357).
* Let $E,F$ be nuclear Frechet spaces. In the following equalities, $\Hom$-spaces between nuclear spaces will be equipped with the topology of uniform convergence on bounded subsets, as above. Then, as we see from Treves, page 525:
$$
\begin{align*}
E \otimes F &= \Hom ( E^\vee, F) \\
E^\vee \otimes F &= \Hom ( E , F) \\
E^\vee \otimes F^\vee &= (E \otimes F)^\vee\\
E^\vee \otimes F^\vee &= \Hom(E,F^\vee)
\end{align*}
$$
* A slightly stronger equation of this nature (from Treves, page 522, proposition 50.5) is the following. If $E$ is of type ($\mc{LF}$) and $F$ is any nuclear space, then
$$
E^\vee \otimes F = \Hom(E,F).
$$
* Finally, for any nuclear spaces $E$ and $F$, we have the following isomorphism (Treves, page 522, proposition 50.4) //only of vector spaces without topology//
$$
E^\vee \otimes F = \Hom (E,F).
$$
This last isomorphism is, in fact, a topological isomorphism where $\Hom(E,F)$ is equipped with a certain topology which is in general different from that described above.
//Caution//: the evaluation map
$$E^\vee \times E \to \R$$
is not, in general, continuous (even if $E$ is a Frechet space). One can see this easily when $E = \cinfty(M)$, where $M$ is a compact manifold. Then, $E^\vee = \mc{D}(M)$ is the space of supported distributions on $M$. If the evaluation map
$$
\cinfty(M) \times \mc{D}(M) \to \R
$$
were continuous, it would extend to a continuous linear map
$$
\cinfty(M) \otimes \mc{D}(M) = \Hom(\cinfty(M), \cinfty(M)) \to \R.
$$
On the subspace
$$
\cinfty(M) \otimes \Omega^n(M),
$$
the putative evaluation map must be given by restriction to the diagonal $M \subset M \times M$ and integrating. However, $\cinfty(M) \otimes \mc{D}(M)$ contains the $\delta$-distribution on the diagonal; and we can not restrict this to the diagonal.
This means that, although the spaces of linear maps between nuclear spaces are topological vector spaces, $Nuc$ //does not// form a symmetric monoidal category enriched in topological vector spaces (with the projective tensor product).
!! Why we care
In field theory, we usually focus on [[ action functionals|local action functional]] that are defined on smooth sections of some vector bundle. We might hope to study the [[ critical locus|derived critical locus]] of such an action, in the spirit of algebraic geometry, //as a moduli problem//. This approach tends to foreground structural features of an equation, although it leaves many concrete questions (such as the solutions in some function space) almost untouched. Because $Nuc$ is a well-behaved symmetric monoidal category, we //can// construct (at least formal) algebraic geometry over this category and thus pursue this viewpoint. In fact, the classical BV formalism seems to be best understood as a construction in derived algebraic geometry over $Nuc$: we describe a field theory with constraints (e.g., gauge theories) as a chain complex of free theories.
Moreover, perturbative quantum field theory fits nicely into this framework, since Wick's lemma and Feynman diagram methods can also be phrased in purely algebraic terms (see "Sums over graphs," IV.3 in \cite{Manin}). When combined with the [[local-effective correspondence]], we have a formalism that captures mathematically much of perturbative QFT.
We work with field theory as developed in \cite{webbook}, which encompasses essentially all field theories studied in physics. The introductory chapter of \cite{webbook} is a leisurely exposition of the main physical and mathematical ideas, and we encourage the reader to examine it before delving into this site. The approach is perturbative, and hence has the flavor of formal geometry. We hope to treat the global version in future work. Here we emphasize two main themes important to our main theorem.
In \cite{webbook}, it is shown how to make mathematical sense of perturbative computations of the path integral, and we use many results found there. Two principles of \cite{webbook} play an important role in our constructions here. The first principle fits into a general trend in recent algebraic geometry, namely that the functorial approach to geometry works over any well-behaved symmetric monoidal category. In our situation, we find that the natural, mathematical home for perturbative quantum field theory is algebraic geometry over the [[symmetric monoidal category of nuclear spaces|overview of nuclear spaces]]. More precisely, relative algebraic geometry over the category of nuclear spaces is adequate for scalar and vector field theories, but for gauge theories (and for a better viewpoint in general) we work over the category of dg nuclear spaces, a setting that might be known as "derived geometry" for field theory. What makes this language appealing is that many procedures in physics, which have rigorous finite-dimensional versions, can be carried out in infinite dimensions using nuclear spaces. In addition, this language provides simple geometric interpretations for these procedures.
The second principle comes from physics, namely that effective field theory provides the right conceptual framework for renormalization. In \cite{webbook} this principle is codified in the [[local-effective correspondence]]. We gloss this correspondence as follows. A [[ perturbative field theory|quantum field theory]] is defined to be a family of effective field theories parametrized by length scale, where the effective theory at a length scale $L$ is obtained from the effective theory at scale $\epsilon$ by integrating out over fields with length scale between $\epsilon$ and $L$. A [[local action functional]] $S$ is a real-valued function on the space of fields where $S(\phi)$ is given by integrating some function of the field and its derivatives over the base manifold (the "spacetime"). The local-effective correspondence is a bijective correspondence between the moduli space of local action functionals and the moduli space of perturbative field theories. Although the starting point for many physical constructions is a local action, one cannot naively apply these constructions to such an action. But, as shown in \cite{webbook}, these constructions //do// work for effective field theories. Hence the local-effective correspondence allows us to rigorously enact these procedures by defining the answer for the local side as the answer given on the effective side.
Recall that the collection of observables in quantum mechanics form an associative algebra. The observables of a classical mechanical system form a Poisson algebra. In the deformation quantization approach to quantum mechanics, one starts with a Poisson algebra $A^{cl}$, and attempts to construct an associative algebra $A^q$, which is an algebra flat over the ring $\mathbb C[[\hbar]] $, together with an isomorphism of associative algebras $A^q / \hbar \cong A^{cl}$. In addition, if $a,b \in A^{cl}$, and $\widetilde{a}, \widetilde{b}$ are any lifts of $a,b$ to $A^q$, then
$$
\lim_{\hbar \to 0} \frac{1}{\hbar} [\widetilde{a}, \widetilde{b} ] = \{a,b\} \in A^{cl}.
$$
This paper concerns the analog, in quantum field theory, of the deformation quantization picture in quantum mechanics. We have seen that the sheaf of solutions to the ~Euler-Lagrange equation of a classical field theory can be encoded by a [[commutative factorization algebra]]. A commutative factorization algebra is the analog, in our setting, of the commutative algebra appearing in deformation quantization. We have argued that the observables of a quantum field theory should form a [[factorization algebra]]. This factorization algebra is the analog of the associative algebra appearing in deformation quantization.
In deformation quantization, the commutative algebra of classical observables has an extra structure -- a Poisson bracket -- which makes it "want" to deform into an associative algebra. In this section we will explain the analogous structure on a commutative factorization algebra which makes it want to deform into a factorization algebra. [[ Later|P 0 Structure Classical Field Theory]] we will see that the commutative factorization algebra associated to a classical field theory has this extra structure.
!!! The $E_0$ operad
''Definition''
//Let $E_0$ be the operad defined by//
$$
E_0 (n) = \begin{cases} 0 &\text { if } n > 0\\
\mathbb{R} &\text{ if } n = 0
\end{cases}
$$
Thus, an $ E_0 $ algebra in the category of real vector spaces is a real vector space with a distinguished element in it. More generally, an $ E_0 $ algebra in a symmetric monoidal category $ \mathcal C $ is the same thing as an object $ A $ of $ \mathcal C $ together with a map $ 1_{\mathcal C} \rightarrow A $
The reason for the terminology $ E_0 $ is that this operad can be interpreted as the operad of little $ 0 $-discs.
The inclusion of the empty set into every open set implies that, for any factorization algebra $ \mathcal F $, there is a unique map from the unit factorization algebra $ \mathbb R \rightarrow \mathcal F $.
!!! The $ P_0 $ operad
The Poisson operad is an object interpolating between the commutative operad and the associative (or $E_1$) operad. We would like to find an analog of the Poisson operad which interpolates between the commutative operad and the $E_0$ operad.
Let us define the Poisson_k (or $P_k$) operad to be the operad whose algebras are commutative algebras equipped with a Poisson bracket of degree $1-k$. With this notation, the usual Poisson operad is the Poisson_1 operad.
Recall that the homology of the $E_n$ operad is the $P_n$ operad, for $n > 1$. Thus, just as the semi-classical version of an algebra over the $E_1$ operad is a Poisson algebra in the usual sense (that is, a $P_1$ algebra), the semi-classical version of an $E_n$ algebra is a $P_n$ algebra.
Thus, we have the following table:
| Classical | Quantum |
| ? | $E_0$ operad |
| $P_1$ operad | $E_1$ operad |
| $P_2$ operad | $E_2$ operad |
| $\vdots$ | $\vdots$ |
This immediately suggests that the $P_0$ operad is the semi-classical version of the $E_0$ operad.
Note that the $P_0$ operad is a Hopf operad: the coproduct is defined by
$$
\begin{split}
c (\star ) &= \star \otimes \star \\
c (\{-,-\} ) &= \{-,-\} \otimes \star + \star \otimes \{-,-\}.
\end{split}
$$
In concrete terms, this means that if $A$ and $B$ are $P_0$ algebras, their tensor product $A \otimes B$ is again a $P_0$ algebra, with product and bracket defined by
$$
\begin{split}
(a \otimes b) \star (a' \otimes b') &= (-1)^{\abs{a'} \abs{b}} (a \star a' ) \otimes (b \star b') \\
\{a \otimes b, a' \otimes b'\} &= (-1)^{\abs{a'} \abs{b} }\left( \{a,a'\} \otimes (b \star b') + (a \star a') \otimes \{b, b'\} \right) .
\end{split}
$$
!! Quantization of $P_0$ algebras
We know what it means to quantize an Poisson algebra in the ordinary sense (that is, a $P_1$ algebra) into an $E_1$ algebra.
There is a similar notion of quantization for $P_0$ algebras. A quantization is simply an $E_0$ algebra over $\R[[\hbar]]$ which, modulo $\hbar$, is the original $P_0$ algebra, and for which there is a certain compatibility between the Poisson bracket on the $P_0$ algebra and the quantized $E_0$ algebra.
Let $A$ be a commutative algebra in the category of cochain complexes. Let $A_1$ be an $E_0$ algebra flat over $\R[[\hbar]]/ \hbar^2$, and suppose that we have an isomorphism of chain complexes
$$
A_1 \otimes_{\R[[\hbar]]/ \hbar^2} \R \cong A.
$$
In this situation, we can define a bracket on $A$ of degree $1$, as follows.
We have an exact sequence
$$
0 \to \hbar A \to A_1 \to A \to 0.
$$
The boundary map of this exact sequence is a cochain map
$$
D : A \to A
$$
(well-defined up to homotopy).
Let us define a bracket on $A$ by the formula
$$
\{a,b\} = D ( a b ) - (-1)^{\abs{a}} a D b - (D a ) b.
$$
Because $D$ is well-defined up to homotopy, so is this bracket. However, unless $D$ is an order two differential operator, this bracket is simply a cochain map $A \otimes A \to A$, and not a Poisson bracket of degree $1$.
In particular, this bracket induces one on the cohomology $H^\ast(A)$ of $A$. The cohomological bracket is independent of any choices.
''Definition.''
//Let $A$ be a $P_0$ algebra in the category of cochain complexes. Then a quantization of $A$ is an $E_0$ algebra $\til{A}$ over $\R[[\hbar]]$, together with a quasi-isomorphism of $E_0$ algebras//
$$
\til{A} \otimes_{\R[[\hbar]]} \R \iso A,
$$
//which satisfies the following //correspondence principle//: the bracket on $H^\ast(A)$ induced by $\til{A}$ must coincide with that given by the $P_0$ structure on $A$.//
[[Later| Beilinson-Drinfeld operad]], we will consider a more sophisticated, operadic notion of quantization, which is strictly stronger than this one. To distinguish between the two notions, one could call the definition of quantization presented here a //weak quantization//, while the definition introduced later will be called a //strong quantization//.
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When we construct the factorization algebra of observables of a quantum field theory, we need to use a more general version of the renormalization group flow based on a general class of parametrices for the operator $[Q,\GF]$. The RG flow based on the heat kernel will arise as a special case. In this page, we will introduce the concept of parametrix.
Recall that our space of fields on $M$ is the space of sections of a graded vector bundle $E$. We will let $\br{\E}(U)$ denote the space of distributional sections of $E$ on $U$, and $\br{\E}_c(U)$ denote the compactly supported distributional sections.
The heat kernel $K_0$ at time zero is an element of $\br{\E}(U) \otimes \br{\E}(U)$, that is, a distributional section of $E \boxtimes E$ on $U \times U$ (as always, tensor products are completed).
Before we define the notion of parametrix, we need a technical definition.
''Definition.'' //If $M$ is a manifold, a subset $V \subset M^n$ is //proper// if all of the projection maps $\pi_1,\ldots,\pi_n : V \to M$ are proper. We say that a function, distribution etc. on $M^n$ has proper support if its support is a proper subset of $M^n$.//
''Definition.'' //A parametrix// $\Phi$ //is a distributional section//
$$
\Phi \in \br{\E}(M) \otimes \br{\E}(M)
$$
//of the bundle// $E \boxtimes E$ //on// $M \times M$, //with the properties that//
# $\Phi$ //is symmetric under the natural// $\Z/2$ //action on// $\br{\E}(M) \otimes \br{\E}(M)$.
# $\Phi$// is of cohomological degree// $1$.
# $\Phi$ //is closed under the natural differential// $Q \otimes 1 + 1 \otimes Q$ //on// $\br{\E}(M) \otimes \br{\E}(M)$.
# $\Phi$ //has proper support.//
# //Let// $\GF : \E \to \E$ //be the gauge fixing operator. We require that//
$$
([Q, \GF] \otimes 1) \Phi - K_0
$$
//is a smooth section of// $E \boxtimes E$// on //$M \times M$.// Thus,//
$$
([Q, \GF] \otimes 1) \Phi - K_0 \in \E(M) \otimes \E(M).
$$
Note the following.
# The kernel $\int_{0}^L K_t \d t$ is a parametrix, for any $L$.
# If $\Phi, \Psi$ are parametrices, then $\Phi - \Psi$ is smooth. Indeed, $\Phi - \Psi$ is in the kernel of the elliptic operator $([Q,\GF]) \otimes 1 + 1 \otimes ([Q,\GF])$ on $\E(M) \otimes \E(M)$.
If $\Phi, \Psi$ are parametrices, we say that $\Phi < \Psi$ if the support of $\Phi$ is contained in the support of $\Psi$. In this way, parametrices acquire a partial order.
//Remark:// For clarity's sake, note that we are defining here parametrices for the generalized Laplacian $[Q, Q^{GF}]$, not parametrices in general.
!!! The propagator for a parametrix
If $\Phi$ is a parametrix, we let
$$
P(\Phi) = (\GF \otimes 1) \Phi.
$$
This is the propagator associated to $\Phi$. We let
$$
K_\Phi = K_0 - ([Q,\GF] \otimes 1) \Phi.
$$
Note that if
$$
\Phi = \int_{0}^L K_t \d t
$$
then
$$
P(\Phi ) = P(0,L) = \int_0^L (\GF \otimes 1) K_t \d t
$$
and
$$
K_\Phi = K_L.
$$
/***
|''Name:''|PasswordOptionPlugin|
|''Description:''|Extends TiddlyWiki options with non encrypted password option.|
|''Version:''|1.0.2|
|''Date:''|Apr 19, 2007|
|''Source:''|http://tiddlywiki.bidix.info/#PasswordOptionPlugin|
|''Author:''|BidiX (BidiX (at) bidix (dot) info)|
|''License:''|[[BSD open source license|http://tiddlywiki.bidix.info/#%5B%5BBSD%20open%20source%20license%5D%5D ]]|
|''~CoreVersion:''|2.2.0 (Beta 5)|
***/
//{{{
version.extensions.PasswordOptionPlugin = {
major: 1, minor: 0, revision: 2,
date: new Date("Apr 19, 2007"),
source: 'http://tiddlywiki.bidix.info/#PasswordOptionPlugin',
author: 'BidiX (BidiX (at) bidix (dot) info',
license: '[[BSD open source license|http://tiddlywiki.bidix.info/#%5B%5BBSD%20open%20source%20license%5D%5D]]',
coreVersion: '2.2.0 (Beta 5)'
};
config.macros.option.passwordCheckboxLabel = "Save this password on this computer";
config.macros.option.passwordInputType = "password"; // password | text
setStylesheet(".pasOptionInput {width: 11em;}\n","passwordInputTypeStyle");
merge(config.macros.option.types, {
'pas': {
elementType: "input",
valueField: "value",
eventName: "onkeyup",
className: "pasOptionInput",
typeValue: config.macros.option.passwordInputType,
create: function(place,type,opt,className,desc) {
// password field
config.macros.option.genericCreate(place,'pas',opt,className,desc);
// checkbox linked with this password "save this password on this computer"
config.macros.option.genericCreate(place,'chk','chk'+opt,className,desc);
// text savePasswordCheckboxLabel
place.appendChild(document.createTextNode(config.macros.option.passwordCheckboxLabel));
},
onChange: config.macros.option.genericOnChange
}
});
merge(config.optionHandlers['chk'], {
get: function(name) {
// is there an option linked with this chk ?
var opt = name.substr(3);
if (config.options[opt])
saveOptionCookie(opt);
return config.options[name] ? "true" : "false";
}
});
merge(config.optionHandlers, {
'pas': {
get: function(name) {
if (config.options["chk"+name]) {
return encodeCookie(config.options[name].toString());
} else {
return "";
}
},
set: function(name,value) {config.options[name] = decodeCookie(value);}
}
});
// need to reload options to load passwordOptions
loadOptionsCookie();
/*
if (!config.options['pasPassword'])
config.options['pasPassword'] = '';
merge(config.optionsDesc,{
pasPassword: "Test password"
});
*/
//}}}
/***
|Name|Plugin: jsMath|
|Created by|BobMcElrath|
|Email|my first name at my last name dot org|
|Location|http://bob.mcelrath.org/tiddlyjsmath.html|
|Version|1.5.1|
|Requires|[[TiddlyWiki|http://www.tiddlywiki.com]] ≥ 2.0.3, [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]] ≥ 3.0|
!Description
LaTeX is the world standard for specifying, typesetting, and communicating mathematics among scientists, engineers, and mathematicians. For more information about LaTeX itself, visit the [[LaTeX Project|http://www.latex-project.org/]]. This plugin typesets math using [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], which is an implementation of the TeX math rules and typesetting in javascript, for your browser. Notice the small button in the lower right corner which opens its control panel.
!Installation
In addition to this plugin, you must also [[install jsMath|http://www.math.union.edu/~dpvc/jsMath/download/jsMath.html]] on the same server as your TiddlyWiki html file. If you're using TiddlyWiki without a web server, then the jsMath directory must be placed in the same location as the TiddlyWiki html file.
I also recommend modifying your StyleSheet use serif fonts that are slightly larger than normal, so that the math matches surrounding text, and \\small fonts are not unreadable (as in exponents and subscripts).
{{{
.viewer {
line-height: 125%;
font-family: serif;
font-size: 12pt;
}
}}}
If you had used a previous version of [[Plugin: jsMath]], it is no longer necessary to edit the main tiddlywiki.html file to add the jsMath <script> tag. [[Plugin: jsMath]] now uses ajax to load jsMath.
!History
* 11-Nov-05, version 1.0, Initial release
* 22-Jan-06, version 1.1, updated for ~TW2.0, tested with jsMath 3.1, editing tiddlywiki.html by hand is no longer necessary.
* 24-Jan-06, version 1.2, fixes for Safari, Konqueror
* 27-Jan-06, version 1.3, improved error handling, detect if ajax was already defined (used by ZiddlyWiki)
* 12-Jul-06, version 1.4, fixed problem with not finding image fonts
* 26-Feb-07, version 1.5, fixed problem with Mozilla "unterminated character class".
* 27-Feb-07, version 1.5.1, Runs compatibly with TW 2.1.0+, by Bram Chen
!Examples
|!Source|!Output|h
|{{{The variable $x$ is real.}}}|The variable $x$ is real.|
|{{{The variable \(y\) is complex.}}}|The variable \(y\) is complex.|
|{{{This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.}}}|This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.|
|{{{This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.}}}|This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.|
|{{{Block formatted equations may also use the 'equation' environment \begin{equation} \int \tan x = -\ln \cos x \end{equation} }}}|Block formatted equations may also use the 'equation' environment \begin{equation} \int \tan x = -\ln \cos x \end{equation}|
|{{{Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} }}}|Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} |
|{{{I spent \$7.38 on lunch.}}}|I spent \$7.38 on lunch.|
|{{{I had to insert a backslash (\\) into my document}}}|I had to insert a backslash (\\) into my document|
!Code
***/
//{{{
// AJAX code adapted from http://timmorgan.org/mini
// This is already loaded by ziddlywiki...
if(typeof(window["ajax"]) == "undefined") {
ajax = {
x: function(){try{return new ActiveXObject('Msxml2.XMLHTTP')}catch(e){try{return new ActiveXObject('Microsoft.XMLHTTP')}catch(e){return new XMLHttpRequest()}}},
gets: function(url){var x=ajax.x();x.open('GET',url,false);x.send(null);return x.responseText}
}
}
// Load jsMath
jsMath = {
Setup: {inited: 1}, // don't run jsMath.Setup.Body() yet
Autoload: {root: new String(document.location).replace(/[^\/]*$/,'jsMath/')} // URL to jsMath directory, change if necessary
};
var jsMathstr;
try {
jsMathstr = ajax.gets(jsMath.Autoload.root+"jsMath.js");
} catch(e) {
alert("jsMath was not found: you must place the 'jsMath' directory in the same place as this file. "
+"The error was:\n"+e.name+": "+e.message);
throw(e); // abort eval
}
try {
window.eval(jsMathstr);
} catch(e) {
alert("jsMath failed to load. The error was:\n"+e.name + ": " + e.message + " on line " + e.lineNumber);
}
jsMath.Setup.inited=0; // allow jsMath.Setup.Body() to run again
// Define wikifers for latex
config.formatterHelpers.mathFormatHelper = function(w) {
var e = document.createElement(this.element);
e.className = this.className;
var endRegExp = new RegExp(this.terminator, "mg");
endRegExp.lastIndex = w.matchStart+w.matchLength;
var matched = endRegExp.exec(w.source);
if(matched) {
var txt = w.source.substr(w.matchStart+w.matchLength,
matched.index-w.matchStart-w.matchLength);
if(this.keepdelim) {
txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
}
e.appendChild(document.createTextNode(txt));
w.output.appendChild(e);
w.nextMatch = endRegExp.lastIndex;
}
}
config.formatters.push({
name: "displayMath1",
match: "\\\$\\\$",
terminator: "\\\$\\\$\\n?", // 2.0 compatability
termRegExp: "\\\$\\\$\\n?",
element: "div",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
config.formatters.push({
name: "inlineMath1",
match: "\\\$",
terminator: "\\\$", // 2.0 compatability
termRegExp: "\\\$",
element: "span",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
var backslashformatters = new Array(0);
backslashformatters.push({
name: "inlineMath2",
match: "\\\\\\\(",
terminator: "\\\\\\\)", // 2.0 compatability
termRegExp: "\\\\\\\)",
element: "span",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
backslashformatters.push({
name: "displayMath2",
match: "\\\\\\\[",
terminator: "\\\\\\\]\\n?", // 2.0 compatability
termRegExp: "\\\\\\\]\\n?",
element: "div",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
backslashformatters.push({
name: "displayMath3",
match: "\\\\begin\\{equation\\}",
terminator: "\\\\end\\{equation\\}\\n?", // 2.0 compatability
termRegExp: "\\\\end\\{equation\\}\\n?",
element: "div",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
// These can be nested. e.g. \begin{equation} \begin{array}{ccc} \begin{array}{ccc} ...
backslashformatters.push({
name: "displayMath4",
match: "\\\\begin\\{eqnarray\\}",
terminator: "\\\\end\\{eqnarray\\}\\n?", // 2.0 compatability
termRegExp: "\\\\end\\{eqnarray\\}\\n?",
element: "div",
className: "math",
keepdelim: true,
handler: config.formatterHelpers.mathFormatHelper
});
// The escape must come between backslash formatters and regular ones.
// So any latex-like \commands must be added to the beginning of
// backslashformatters here.
backslashformatters.push({
name: "escape",
match: "\\\\.",
handler: function(w) {
w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));
w.nextMatch = w.matchStart+2;
}
});
config.formatters=backslashformatters.concat(config.formatters);
window.wikify = function(source,output,highlightRegExp,tiddler)
{
if(source && source != "") {
if(version.major == 2 && version.minor > 0) {
var wikifier = new Wikifier(source,getParser(tiddler),highlightRegExp,tiddler);
wikifier.subWikifyUnterm(output);
} else {
var wikifier = new Wikifier(source,formatter,highlightRegExp,tiddler);
wikifier.subWikify(output,null);
}
jsMath.ProcessBeforeShowing();
}
}
//}}}
Let $M$ be a topological space and let $(C, \otimes)$ be a symmetric monoidal category. We are particularly interested in the case where $M$ is a smooth manifold and $C$ is $\op{Vect}$ or $\op{dgVect}$ with the usual tensor product as the monoidal product.
!!! The essential idea of a prefactorization algebra
A prefactorization algebra $\mathcal{F}$ on $M$, taking values in cochain complexes, is a rule that assigns a cochain complex $\mathcal{F}(U)$ to each open set $U \subset M$ along with
* a cochain map $\mathcal{F}(U) \rightarrow \mathcal{F}(V)$ for each inclusion $U \subset V$;
* a cochain map $\mathcal{F}(U_1) \otimes \cdots \otimes \mathcal{F}(U_n) \rightarrow \mathcal{F}(V)$ for every finite collection of open sets where each $U_i \subset V$ and the $U_i$ are pairwise disjoint;
* the maps are compatible in the obvious way (e.g. if $U \subset V \subset W$ is a sequence of open sets, the map $\mathcal{F}(U) \rightarrow \mathcal{F}(W)$ agrees with the composition through $\mathcal{F}(V)$).
It resembles a precosheaf, except that we tensor the cochain complexes rather than taking their direct sum.
!!! Prefactorization algebras in the style of algebras over an operad
''Definition'' Let $\operatorname{Fact}_M$ denote the following multicategory associated to $M$.
* The objects consist of all connected open subsets of $M$.
* For every (possibly empty) finite collection of open sets $\{U_\alpha\}_{\alpha \in A}$ and open set $V$, there is a set of maps $\operatorname{Fact}_M( \{U_\alpha\}_{\alpha \in A}, V)$. If the $U_\alpha$ are pairwise disjoint and all are contained in $V$, then the set of maps is a single point. Otherwise, the set of maps is empty.
* The composition of maps is defined in the obvious way.
//Remark:// By 'multicategory' we mean what is also called a colored operad or a pseudo-tensor category. We use Leinster's definition of a 'fat symmetric multicategory'.
Recall that any symmetric monoidal category $C$ gives rise to a canonical multicategory $\hat{C}$. Namely, for any finite collection $\{a_1,\ldots,a_n\}$ of objects in $C$ and any object $b$ in $C$, the set of maps $\hat{C}(\{a_1,\ldots,a_n\}, b)$ in the multicategory is exactly the set of maps $C(a_1 \otimes \cdots \otimes a_n, b)$ in the category $C$. We will sometimes identify both the symmetric monoidal category and its associated multicategory by the same symbol.
''Definition'' A //prefactorization algebra// on $M$ taking values in $C$ is a functor (of multicategories) from $\operatorname{Fact}_M$ to $\hat{C}$.
//Remark:// In other words, a prefactorization algebra //is// an algebra over the colored operad $\operatorname{Fact}_M$.
//Remark:// When the monoidal product on $C$ is the coproduct, then a precosheaf on $M$ defines a prefactorization algebra. Hence, our definition broadens the idea of "inclusion of open sets leads to inclusion of sections" by allowing more general monoidal structures to "combine" the sections on disjoint open sets. Something analogous happens when we equip the category of abelian groups with the tensor product as its monoidal structure.
!!! Prefactorization algebras in the style of precosheaves
Any multicategory $ \mscr{C} $ has an associated symmetric monoidal category $ S\mscr{C} $, which is defined to be the universal symmetric monoidal category equipped with a functor of multicategories $ \mscr{C} \rightarrow S \mscr{C} $.
We can give an alternative definition of prefactorization algebra by working with the symmetric monoidal category $ S \operatorname{Fact}_M $ rather than the multicategory $ \operatorname{Fact}_M $.
''Definition'' Let $S\operatorname{Fact}_M$ denote the following symmetric monoidal category.
* The objects of $ S \operatorname{Fact}_M $ consist of topological spaces $ U $ equipped with a map $ U \rightarrow M $ which, on each connected component of $ U $, is an open embedding embedding.
* A map from $U \to M$ to $V \to M$ is a commutative diagram
$$
\begin{array}{ccc}
U & \xto{i} & V \\
\downarrow & \swarrow\\
M
\end{array}
$$
where the map $ i $ is an embedding.
* The symmetric monoidal structure on $S \op{Fact}_M$ is given by disjoint union.
''Lemma''. $ S \operatorname{Fact}_M $ is the universal symmetric monoidal category containing the multicategory $ \operatorname{Fact}_M $.
The alternative definition of prefactorization algebra is as follows.
''Definition'' A prefactorization algebra with values in a symmetric monoidal category $ \mathscr{C} $ is a symmetric monoidal functor
$$
S \operatorname{Fact}_M \rightarrow \mathscr{C}.
$$
//Remark:// Although "algebra" appears in its name, a prefactorization algebra only allows one to "multiply" elements that live on disjoint open sets. The category of prefactorization algebras (taking values in some fixed target category) has a symmetric monoidal product, so we can study commutative algebra objects in that category. As an example, we will consider the observables for a [[classical field theory]].
!!! Comparison with $E_n$ algebras
If $M$ is an $n$-dimensional manifold, then prefactorization algebras locally bear a resemblance to $E_n$ algebras. After all, a prefactorization algebra prescribes a way to combine the elements associated to $k$ distinct balls into an element associated to a big ball containing all $k$ balls.
In fact, Lurie \cite{} has shown the following.
''Theorem''. Let $F$ be a prefactorization algebra on $\R^n$ with values in cochain complexes, with the property that for any two balls $B, B'$, the map
$$
F(B) \to F(B')
$$
is a quasi-isomorphism. Then $F(B)$ is an $E_n$ algebra.
The prefactorization algebras considered in this paper will not, in general, have this property.
The (pre)-factorization algebras of interest in this paper arise from perturbative quantum field theories. We have [[already | the motivating example of quantum mechanics]] discussed how factorization algebras appear in quantum mechanics. In this page we will see that this picture extends in a very natural way to quantum field theory.
The manifold $M$ on which the prefactorization algebra is defined is the space-time manifold of the quantum field theory. If $U \subset M$ is an open subset, we will interpret $\mc{F}(U)$ as the space of observables (or measurements) that we can make, which only depend on the behaviour of the fields on $U$. Performing a measurement involves coupling a measuring device to the quantum system in the region $U$.
One can bear in mind the example of a particle accelerator. In that situation, one can imagine the space-time $M$ as being of the form $M = A \times (0,t)$, where $A$ is the interior of the accelerator and $t$ is the duration of our experiment.
In this situation, performing a measurement on some open subset $U \subset M$ is something concrete. Let us take $U = V \times (\eps,\delta)$, where $V \subset A$ is some small region in the accelerator, and $(\eps,\delta)$ is a short time interval. Performing a measurement on $U$ amounts to coupling a measuring device to our accelerator in the region $V$, starting at time $\eps$ and ending at time $\delta$. For example, we could imagine that there is some piece of equipment in the region $V$ of the accelerator, which is switched on at time $\eps$ and switched off at time $\delta$.
!!! Interpretation of the prefactorization algebra axioms
Suppose that we have two different measuring devices, $O_1$ and $O_2$. We would like to set up our accelerator so that we measure both $O_1$ and $O_2$.
There are two ways we can do this. Either we insert $O_1$ and $O_2$ into disjoint regions $V_1, V_2$ of our accelerator. Then we can turn $O_1$ and $O_2$ on at any times we like, including for overlapping time intervals.
If the regions $V_1, V_2$ overlap, then we can not do this. After all, it doesn't make sense to have two different measuring devices at the same point in space at the same time.
However, we could imagine inserting $O_1$ into region $V_1$ during the time interval $(a,b)$; and then removing $O_1$, and inserting $O_2$ into the overlapping region $V_2$ for the disjoint time interval $(c,d)$.
These simple considerations immediately suggest that the possible measurements we can make of our physical system form a prefactorization algebra. Let $\Obs(U)$ denote the space of measurements we can make on an open subset $U \subset M$. Then, by combining measurements in the way outlined above, we would expect to have maps
$$
\Obs(U)\otimes \Obs(U') \to \Obs(V)
$$
whenever $U, U'$ are disjoint open subsets of an open subset $V$. The associativity and commutativity properties of a prefactorization algebra are evident.
!!! The cochain complex of observables
In the approach to quantum field theory considered in this paper, the factorization algebra of observables will be a factorization algebra of cochain complexes. One can ask for the physical meaning of the cochain complex $\Obs(U)$.
It turns out that the "physical" observables will be $H^0 (\Obs(U))$. If $O \in \Obs^0(U)$ is an observable of cohomological degree $0$, then the equation $\d O = 0$ can often be interpreted as saying that $O$ is compatible with the gauge symmetries of the theory. Thus, only those observables $O \in \Obs^0(U)$ which are closed are physically meaningful.
The equivalence relation identifying $O \in \Obs^0(U)$ with $O + \d O'$, where $O' \in \Obs^{-1}(U)$, also has a physical interpretation, which will take a little more work to describe. Often, two observables on $U$ are physically indistinguishable (that is, they can not be distinguished by any measurement one can perform). In the example of an accelerator outlined above, two measuring devices are equivalent if they always produce the same expectation values, no matter how we prepare our system, or no matter what boundary conditions we impose.
As another example, in the quantum mechanics of a free particle, the observable measuring the momentum of a particle at time $t$ is equivalent to that measuring the momentum of a particle at another time $t'$. This is because, even at the quantum level, momentum is preserved (as the momentum operator commutes with the Hamiltonian).
From the cohomological point of view, if $O, O' \in \Obs^0(U)$ are observables which are in the kernel of $\d$ (and thus "physically meaningful"), then they are equivalent in the sense described above if they differ by an exact observable.
It is a little more difficult to provide a physical interpretation for the non-zero cohomology groups $H^i(\Obs(U))$. The first cohomology group $H^1(\Obs(U))$ is the recipient of any anomalies to defining observables at the quantum level. For example, in a gauge theory, one might have a classical observable which respects gauge symmetry. However, it may not lift to a quantum observable respecting gauge symmetry; this happens if there is a non-zero anomaly in $H^1(\Obs(U))$.
The cohomology groups $H^i(\Obs(U))$, when $i < 0$, are best interpreted as symmetries, and higher symmetries, of observables. Indeed, we have seen that the physically meaningful observables are the closed degree $0$ elements of $\Obs(U)$. One can construct a simplicial set, whose $n$-simplices are closed and degree $0$ elements of $\Obs(U) \otimes \Omega^\ast(\Delta^n)$. The vertices of this simplicial set are observables, the edges are equivalences between observables, the faces are equivalences between equivalences, and so on.
The ~Dold-Kan correspondence tells us that the $i$'th homotopy group of this simplicial set is $H^{-i}(\Obs(U))$. This allows us to interpret $H^{-1}(\Obs(U))$ as being the group of symmetries of the trivial observable $0 \in H^0 (\Obs(U))$, and $H^{-2} (\Obs(U))$ as the symmetries of the identity symmetry of $0 \in H^0 (\Obs(U))$, and so on.
Although the cohomology groups $H^i(\Obs(U))$ where $i > 1$ do not have such a clear physical interpretation, they are mathematically very natural objects and it is important not to discount them.
!!! Local observables
So far, we have defined a BD algebra $\ObsHomotopy(M)$ of global observables which satisfy the renormalization group flow only up to homotopy. Next, we will define the corresponding local observables. This will give the desired BD algebra in factorization algebras.
Recall that we defined
$$
\ObsHomotopy(M) = \Oo(\E(M)) \otimes \A [[\hbar]].
$$
We simply define
$$
\ObsHomotopy(U) = \Oo( \E(U)) \otimes \A [[\hbar]]
$$
to be the subspace of functionals which are supported on $U$.
Recall that every element $O \in \ObsHomotopy(M)$ can be written as
$$
O = \sum \hbar^i O_{i,k}
$$
where
$$
O_{i,k} \in \Hom (\E(M)^{\otimes k}, \R)_{S_k} \otimes \A = \colim_{\delta} \Hom (\E(M)^{\otimes k}, \R)_{S_k} \otimes \Omega^\ast ( (0,\delta)).
$$
The element $O$ is in $\ObsHomotopy^{h}(U)$ if
$$
O_{i,k} \in \Hom (\E(U)^{\otimes k}, \R)_{S_k} \otimes \A = \colim_{\delta} \Hom (\E(U)^{\otimes k}, \R)_{S_k} \otimes \Omega^\ast ( (0,\delta)).
$$
Thus, there exists some $\delta > 0$ so that $O_{i,k}$ lifts to
$$O_{i,k}^\delta \in \Hom (\E(U)^{\otimes k}, \R)_{S_k} \otimes \Omega^\ast ( (0,\delta)).$$
The subspace $\ObsHomotopy(U) \subset \ObsHomotopy(M)$ is clearly a subalgebra of $\ObsHomotopy(M)$. The lemma that makes this definition work is the following.
''Lemma.'' //If $O \in \ObsHomotopy(U)$, then so is $\what{Q}(O)$. Thus, $\ObsHomotopy(U)$ is a subcomplex of $\ObsHomotopy(M)$.//
''Proof.''
It is immediate that $Q, \d_{dR}$ and $\Delta$ preserve $\ObsHomotopy(U)$. By using the fact that $I[\Phi]$ is supported arbitrarily close to the diagonal for sufficiently small parametrices $\Phi$ implies that $\{I,-\}$ preserves $\ObsHomotopy(U)$.
$\square$.
The lemma implies that $\ObsHomotopy(U) \subset \ObsHomotopy(M)$ is also closed under the bracket $\{-,-\}$, and thus forms a sub BD algebra of $\ObsHomotopy(M)$.
!!! Prefactorization structure.
Let $U_1,\ldots, U_n \subset V$ be open subsets of $M$, with $U_i$ disjoint. There is a map
$$
\begin{split}
\ObsHomotopy(U_1) \otimes \cdots \otimes \ObsHomotopy(U_n) & \to \ObsHomotopy(V) \\
\alpha_1 \otimes \cdots \otimes \alpha_n &\to \alpha_1 \cdots \alpha_n
\end{split}
$$
(where $\alpha_1 \cdots \alpha_n$ is simply the product in $\ObsHomotopy(V)$).
''Lemma.'' //This is a cochain map.//
''Proof.'' It suffices to take $n = 2$. Let $\alpha \in \ObsHomotopy(U_1)$ and $\beta \in \ObsHomotopy(U_2)$. We need to show that
$$
\what{Q} (\alpha_1 \alpha_2) = (\what{Q}\alpha_1) \alpha_2 + (-1)^{\abs{\alpha_1}} \alpha_1 \what{Q} \alpha_2.
$$
Recall that
$$
\what{Q} = Q + \d_{dR} + \{I,-\} + \Delta.
$$
All terms in this expression are derivations, except $\Delta$. The failure of $\Delta$ to be a derivation is measured by the bracket $\{-,-\}$. Thus, we have to show that
$$
\{\alpha ,\beta\} = 0.
$$
Let us choose $(i,k)$ and $(r,s)$ in $\Z_{\ge 0} \times \Z_{\ge 0}$. Let
$$
\begin{split}
\alpha_{i,k} &\in \Hom(\E(U_1)^{\otimes k}, \R)_{S_k} \otimes \A \\
\beta_{r,s} &\in \Hom(\E(U_2)^{\otimes s}, \R)_{S_s} \otimes \A.
\end{split}
$$
Now, there exists $\delta > 0$ so that $\alpha_{i,k}$ lifts to
$$
\alpha_{i,k}^\delta \in \Hom (\E(U_1)^{\otimes k}, \R)_{S_k} \otimes \Omega^\ast ( (0,\delta ))
$$
and there is a similar lift of $\beta_{r,s}^\delta$ of $\beta_{r,s}$.
Now,
$$
\{\alpha^\delta_{i,k},\beta^\delta_{r,s}\} \in \Hom ( \E(U_1)^{\otimes k - 1} \otimes \E(U_2)^{\otimes s - 1}, \R)_{S_{k+s-2}} \otimes \Omega^\ast( (0,\delta) )
$$
is defined by contracting one factor of $\alpha_{i,k}$ with one of $\beta_{r,s}$ using a kernel supported within $\delta$ of the diagonal on $M \times M$.
It follows immediately that, for sufficiently small $\delta$, $\{\alpha^\delta_{i,k}, \beta^\delta_{r,s} \} = 0$. This implies that $\{\alpha,\beta\} = 0$, as desired.
$\square$
The fact that the map
$$
\ObsHomotopy(U_1) \otimes \cdots \otimes \ObsHomotopy(U_n) \to \ObsHomotopy(V)
$$
is both a cochain map and a map of commutative algebras implies that it is a map of BD algebras (because the bracket $\{-,-\}$ is defined using the differential). In addition, this map is evidently linear over $\A[[\hbar]]$.
Thus, we have shown the following.
''Proposition''. // Sending $U \to \ObsHomotopy(U)$ defines a BD algebra in prefactorization algebras on $M$, over $\A[[\hbar]]$. //
''Lemma''.
The natural map
$$
\check{C}(\mf{W}, i_\ast^{\mf{U}}(\F)) \to \check{C}(\mf{U}_{\mf{W}}, \F).
$$
is a quasi-isomorphism.
''Proof.''
Before we check this, let us recall the notation we used when discussing Cech complexes. Let $P \mf{U}$ denote the set of subsets $\alpha \subset \mf{U}$, where for each distinct $i,j \in \alpha$, $U_i$ and $U_j$ are disjoint. If $\alpha \in P \mf{U}$ we will let
$$
U_\alpha = \amalg_{i \in \alpha} U_i.
$$
If $\alpha_1,\ldots,\alpha_k \in P \mf{U}$, we will let
$$
\F( \alpha_1,\ldots,\alpha_k) = \oplus_{i_1 \in \alpha_1,\dots, i_k \in \alpha_k } \F( U_{i_1} \cap \cdots \cap U_{i_k} ).
$$
With this notation, if $W \subset M$, then
$$
i_\ast^{\mf{U}}(\F) (W) = \oplus_{\alpha_1,\ldots, \alpha_r \in \mf{U}_W} \F(\alpha_1,\ldots,\alpha_r) [r-1]
$$
where $\mf{U}_W$ refers to the cover of $W$ consisting of open sets in $\mf{U}$ which lie in $W$.
Let us define a filtration on $i_\ast^{\mf{U}}(\F)$ by saying that
$$
F^i i_\ast^{\mf{U}}(\F) = \oplus_{r \le i} \oplus_{\alpha_1,\ldots, \alpha_r \in \mf{U}_W} \F(\alpha_1,\ldots,\alpha_r) [r-1].
$$
This filters $i_\ast^{\mf{U}}(\F)$ as a prefactorization algebra.
There is a natural map
$$
\check{C}(\mf{W}, i_\ast^{\mf{U}}(\F)) \to \check{C}(\mf{U}_{\mf{W}}, \F).
$$
Let us filter $\check{C}(\mf{W}, i_\ast^{\mf{U}}(\F)) $ by the filtration coming from $i_\ast^{\mf{U}}(\F)$. Let us filter $\check{C}(\mf{U}_{\mf{W}}, \F)$ in the same way that we filtered $i_\ast^{\mf{U}}(\F)$. The map preserves the filtration.
Thus, to prove that this map is a quasi-isomorphism, it suffices to show that it is on the associated graded.
The complex $\Gr^n \check{C}(\mf{W}, i_\ast^{\mf{U}}(\F))$ breaks up as a direct sum of pieces corresponding to tuples $\alpha_1,\ldots,\alpha_n \in P \mf{U}_{\mf{W}}$, as follows. If $\beta \in P \mf{W}$ and $\alpha \in P \mf{U}_{\mf{W}}$, say $\alpha \subset \beta$ if $U_{\alpha} \subset U'_\beta$. Then,
$$
\Gr^n \check{C}(\mf{W}, i_\ast^{\mf{U}}(\F)) = \oplus_{\alpha_1,\ldots,\alpha_n \in P \mf{U} } \F(\alpha_1,\ldots,\alpha_n)[n-1] \otimes \left( \oplus_{\substack{\beta_1,\ldots,\beta_m \in P \mf{W} \\ \alpha_i \subset \beta_j \text{ all } i,j}} \C \cdot (\beta_1,\ldots,\beta_k) \right)
$$
Here $(\beta_1,\ldots,\beta_k)$ denotes a vector in degree $-k$. This is a direct sum decomposition of cochain complexes.
On the other hand,
$$
\Gr^n \check{C}(\mf{U}_{\mf{W}}, \F) = \oplus_{\alpha_1,\ldots,\alpha_n \in P \mf{U}_{\mf{W}} } \F( \alpha_1,\ldots,\alpha_n)
$$
Thus, to prove the lemma, we need to verify that the complex
$$
\oplus_{\substack{\beta_1,\ldots,\beta_m \in P \mf{W} \\ \alpha_i \subset \beta_j \text{ all } i,j}} \C \cdot (\beta_1,\ldots,\beta_k)
$$
has homology $\C$ if all $\alpha_i \in P \mf{U}_{\mf{W}}$, and zero otherwise.
It is clear that the complex is zero if all $\alpha_i$ are not in $P \mf{U}_{\mf{W}}$. So let us assume that all $\alpha_i$ are in $P \mf{U}_{\mf{W}}$. Then, the complex is simply the simplicial chain complex on the infinite simplex with vertices $\beta \in P \mf{U}$ such that $\cup U_{\alpha_i} \subset U_\beta$. This is of course contractible.
$\square$
''Lemma.'' //Let $P$ be any fully subcategory of the category of topological vector spaces which is closed under countable products. Let $E$ be a topological cochain complex equipped with a topologically split complete decreasing filtration $F^i E$. Suppose that $\Gr^i E$ is a homotopy $P$-complex. Then so is $E$. //
''Proof.''
Let $F^i$ be a sequence of $P$-spaces equipped with homotopy equivalences
$$
\Gr^i E \simeq F^i.
$$
Let
$$
F = \prod F^i.
$$
There is an isomorphism of graded topological vector spaces
$$
\Gr^\ast E \defeq \prod \Gr^i E \iso E.
$$
Let $\d^E_0$ denote the differential on $E$ arising from the identification $E \iso \Gr^\ast E$. Thus, $\d^E_0$ preserves the grading on $E$ induced from that on $\Gr^\ast E$. Let $\d^E$ denote the usual differential on $E$.
Let $\alpha = \d^E - \d^E_0$. Note that
$$
\alpha \in F^1 \Hom (E, E)
$$
(where we are using the notation as in the proof of the previous lemma). The fact that $(\d^E)^2 = 0$ implies that
$$
\d_0 \alpha + \tfrac{1}{2} [\alpha,\alpha] = 0
$$
so that $\alpha$ satisfies the ~Maurer-Cartan equation.
Let $\d_0^F$ be the differential on $F$. By assumption, here is a cochain homotopy equivalence $(E, \d_0^E) \simeq (F,\d_0^F)$. Let us denote the maps in this homotopy equivalence by
$$
\begin{split}
p : E &\to F \\
q : F & \to E\\
S_E : q \circ p & \simeq \op{Id} _E \\
S_F : p \circ q & \simeq \op{Id}_F.
\end{split}
$$
Note that the homotopies $S_E$ and $S_F$ preserve the gradings on $E$ and $F$.
Let us define
$$
\beta \in F^1 \Hom(F,F)
$$
by the formula
$$
\beta = \sum_{n \ge 0} q \alpha (S_E \alpha)^n p.
$$
The sum converges, because the filtration on $\Hom(F,F)$ is complete, and $\alpha (S_E \alpha)^n$ is in $F^{n+1} \Hom(F,F)$.
Then, one can check directly that
$$
\d_0^F \beta + \tfrac{1}{2} [\beta,\beta] = 0.
$$
Let
$$
\d^F = \d_0^F + \beta.
$$
Then, $(\d^F)^2 = 0$.
To complete the proof, we need to exhibit a cochain homotopy equivalence between $(E, \d^E)$ and $(F, \d^F)$. By the previous lemma, it suffices to exhibit a filtration preserving cochain map
$$
(F, \d^F) \to (E, \d^E)
$$
which is a homotopy equivalence on the associated graded.
We can write down this map explicitly. Let
$$
\til{q} = \sum_{n \ge 0} (S_E \alpha)^n q.
$$
The sum converges, again because the filtration on $\Hom(F,E)$ is complete, and $(S_E \alpha)^n q$ is in $F^n \Hom(F,E)$.
It is clear that $\Gr^0 \til{q} = q$, so that on the associated graded, $\til{q}$ is a homotopy equivalence.
A direct check shows that $\til{q}$ is a cochain map, thus completing the proof.
$\square$
''Lemma.'' //Let $E$, $G$ be topological cochain complexes equipped with a topologically split complete decreasing filtrations $F^i E$, $F^i G$. Let $f : E \to G$ be a filtration preserving cochain map. Suppose that//
$$
\Gr^i f : \Gr^i E \to \Gr^i G
$$
//is a homotopy equivalence. Then so is $f$.//
''Proof.''
Let us treat the cochain complex $\Hom(E,F)$ of continuous linear maps from $E$ to $F$ just as a cochain complex of vector spaces, not of topological vector spaces. Let
$$
F^k \Hom(E,F) \subset \Hom(E,F)
$$
be the subspace of maps which take $F^i E$ to $F^{i+k} G$, for each $i$. Thus, $F^0 \Hom(E,G)$ is the space of filtration preserving maps.
Note that we have
$$
F^0 \Hom(E,G) = \liminv_k F^0 \Hom (E,G) / F^k \Hom(E,G).
$$
Indeed, $E = \prod \Gr^i E$ as graded topological vector spaces, and similarly for $G$.
$$
\Hom(E,G) = \prod_\Hom(E, \Gr^j G).
$$
Further,
$$
F^0 \Hom(E,G) = \prod_j \prod_{i \le j} \Hom( \Gr^i E, \Gr^j G).
$$
Similarly,
$$
F^0 \Hom(E,G) / F^k \Hom(E,G) = \prod_j \prod_{j - k \le i \le j} \Hom (\Gr^i E, \Gr^j G).
$$
From this it is clear that
$$
F^0 \Hom(E,G) = \liminv_k F^0 \Hom (E,G) / F^k \Hom(E,G),
$$
and that further, the maps in the inverse system are surjective cochain maps.
Similar statements hold for $\Hom(G,E)$, $\Hom(E,E)$ and $\Hom(G,G)$.
Now, by assumption, we have $f \in F^0 \Hom(E,G))$ and $g_0 \in \Gr^0 \Hom(G,E)$, together with homotopies
$$
\begin{split}
(\Gr^0 f) \circ g_0 & \simeq \Gr^0 \op{Id} \in \Gr^0 \Hom (G,G) \\
g_0 \circ (\Gr^0 f) & \simeq \Gr^0 \op{Id} \in \Gr^0 \Hom (E,E).
\end{split}
$$
The existence of such homotopies implies that the map
$$
\begin{split}
\Gr^i \Hom(G,E) &\to \Gr^i \Hom(E,E)\\
\phi & \mapsto \phi \circ \Gr^0 f
\end{split}
$$
is a cochain homotopy equivalence (with inverse, of course, given by composing with $g_0$). The same holds for the map
$$
\Gr^i \Hom(G,E) \to \Gr^i \Hom (G,G)
$$
given by postcomposing with $\Gr^0 f$.
There is a short exact sequence of cochain complexes
$$
0 \to \Gr^i \Hom(E,E) \to F^0 \Hom(E,E) / F^{i+1} \Hom(E,E) \to F^0 \Hom(E,E) / F^{i} \Hom(E,E) \to 0
$$
(and similarly for $\Hom(E,G)$, $\Hom(G,E)$ and $\Hom(G,G)$).
By induction, and by considering this short exact sequence, we see that for each $i$, the map
$$
\begin{split}
F^0 \Hom(G,E) / F^i \Hom(G,E) &\to F^0 \Hom(E,E) / F^i \Hom(E,E) \\
\phi & \mapsto f \circ \phi
\end{split}
$$
is a quasi-isomorphism of cochain complexes.
Since $F^0 \Hom(E,E)$ and $F^0 \Hom(G,E)$ both arise as inverse limits of the cochain complexes $F^0 \Hom(E,E) / F^i \Hom(E,E) $ and $F^0 \Hom(G,E) / F^i \Hom(G,E) $, and since the maps in the inverse system are surjective, it follows that the map
$$
\begin{split}
F^0 \Hom(G,E) &\to F^0 \Hom(E,E) \\
\phi & \mapsto \phi \circ f
\end{split}
$$
is a quasi-isomorphism.
Let $g \in F^0 \Hom(G,E)$ be a cochain representative for the class in $H^0 (F^0 \Hom(G,E))$ corresponding to the identity map on $E$. Then, by construction, $g \circ f$ is cochain homotopic to the identity in $E$, so $g$ provides a left homotopy inverse to $f$.
A similar argument shows that the map
$$
\begin{split}
F^0 \Hom(G,E) &\to F^0 \Hom(G,G) \\
\phi & \mapsto f \circ \phi
\end{split}
$$
is a quasi-isomorphism. This allows us to construct a right homotopy inverse for $f$; the usual argument shows that we make take the right and left homotopy inverses to be the same.
$\square$
''Proposition''
Let us choose (arbitrarily) a metric on $M$, and suppose that that $\Phi$ is a parametrix supported within $\delta$ of the diagonal in $M \times M$. Then there is some constant $c_{i,k}$ (independent of $\Phi$ and $\delta$) such that
$I_{i,k}[\Phi]$ is supported within $c_{i,k} \delta$ of the diagonal in $M^k$.
''Proof''
In \cite{webbook} it was shown that we can construct the effective interactions from a local action functional
$$
I \in \mathcal{O}_l^+(\mathcal{E})[[\hbar]]
$$
as a limit
$$
I[L] = \lim_{\eps \to 0} W ( P ( \epsilon, L ) , I - I^{CT}(\epsilon) ).
$$
Here $I^{CT}(\epsilon)$ are the counterterms; they are elements
$$
I^{CT} \in \left( \mathcal{O}_l^+(\mathcal{E}) \otimes_{alg} \cinfty( (0,\infty)_{\epsilon} ) \right)[[\hbar]].
$$
Recall that we are assuming that $\Phi$ is supported within $\delta$ of the diagonal. Let us choose a smooth function $f$ on $M \times M$, which is supported within $\delta$ of the diagonal, and which takes value $1$ in a neighbourhood of the diagonal.
Let
$$
P_f(\eps,L) = f P(\eps,L).
$$
It was shown in \cite{webbook}, Chapter 2, section 12, that the limit
$$
\lim_{\eps \to 0} W( P_f(\eps,L) , I - I^{CT}(\eps))
$$
exists, and that this limit coincides with
$$
W ( P_f(0,L) - P(0,L), I[L] ) .
$$
Let us introduce the notation
$$
P(\eps,\Phi) = P(\Phi) - P_f(0,\eps).
$$
Thus, $P(0,\Phi) = P(\Phi)$. Note that $P(\eps,\Phi)$ is smooth, that is, an element of $\E \otimes \E$. Further, $P(\eps,\Phi)$ is supported within $\delta$ of the diagonal.
''Lemma''. The limit
$$
\lim_{\eps \to 0} W ( P (\eps,\Phi), I - I^{CT}(\eps) )
$$
exists and is equal to
$$
I[\Phi ] = W ( P(\Phi) - P(0,L), I[L] ) .
$$
''Proof.''
Note that
$$
P_f(0,L) - P(\Phi) = P_f(\eps,L) - P(\eps,\Phi)
$$
is also smooth.
Thus,
$$
\begin{split}
W ( P(\eps,\Phi), I - I^{CT}(\eps) ) &= W ( P(\Phi) - P_f(0,L) + P_f(\eps, L), I - I^{CT}(\eps) )\\
&= W ( P(\Phi) - P_f(0,L), W (P_f(\eps,L), I - I^{CT}(\eps) ) ).
\end{split}
$$
Taking the $\eps \to 0$ limit on both sides shows that
$$
\lim_{\eps \to 0} W ( P(\eps,\Phi), I - I^{CT}(\eps) ) = W ( P(\Phi) - P_f(0,L) , I_f[L] )
$$
where
$$
I_f[L] = W ( P_f(0,L) - P(0,L), I[L]).
$$
Thus,
$$
\lim_{\eps \to 0} W ( P(\eps,\Phi), I - I^{CT}(\eps) ) = I[\Phi]
$$
as desired. $\square$
To show the desired estimate about the support of $I_{i,k}[\Phi]$, we only need to show that, for all $0 < \epsilon < 1$, the expression
$$
W_{i,k} ( P(\eps,\Phi) , I - I^{CT}(\epsilon) )
$$
satisfies this property.
To see this, let us recall the Feynman diagram expansion
$$
W_{i,k} ( P(\eps,\Phi) , I - I^{CT}(\epsilon) ) = \sum_\gamma \frac{1}{\abs{Aut}(\gamma)} W_\gamma ( P(\eps,\Phi), I - I^{CT}(\epsilon) )
$$
where the sum is over connected graphs of genus $i$ with $k$ tails, and the weight $ W_\gamma ( P(\eps,\Phi), I - I^{CT}(\epsilon) )$ is defined by contracting the propagator $P(\eps,\Phi)$ with the interaction $I - I^{CT}(\epsilon)$.
The fact that the functional $I$ and the counterterms $I^{CT}(\epsilon)$ are local implies that the support of each $W_\gamma ( P(\eps,\Phi), I - I^{CT}(\epsilon) )$ is within $c_\gamma \delta$ of the small diagonal in $M^k$, where $c_\gamma$ is a constant depending on $\gamma$. Since there are only a finite number of graphs appearing in the sum, the result follows.
Note that
$$\begin{split}
W (P(\Psi) - P(\Gamma), W ( P(\Gamma) - P(\Phi) , I[\Phi] + \delta O[\Phi] \ast O'[\Phi])) \\
= W (P(\Psi) - P(\Phi), I[\Phi] + \delta O[\Phi] \ast O'[\Phi]) .
\end{split}
$$
Thus, to show that the limit
$$
\lim_{\Phi \to 0} W (P(\Psi) - P(\Phi), I[\Phi] + \delta O[\Phi] \ast O'[\Phi])
$$
is eventually constant, it suffices to show that, for all sufficiently small $\Phi,\Gamma$ satisfying $\Phi < \Gamma$,
$$
W_{i,k} (P(\Gamma) - P(\Phi), I[\Phi] + \delta O[\Phi] \ast O'[\Phi]) = I_{i,k}[\Gamma] + \delta (O[\Gamma] \ast O'[\Gamma] )_{i,k}.
$$
This is a simple exercise in the manipulation of Feynman diagrams. Since $\delta^2 = 0$, the weight $w_G$ of a graph $G$ can be written as a sum
\[
w_G(P(\Gamma) - P(\Phi), I[\Phi]) + \delta w^\prime_G,
\]
where the term $w^\prime_G$ arises from the appearance of $\delta O[\Phi] \ast O'[\Phi]$ at some vertex. Thus we see that we can likewise separate the term $W_{i,k} (P(\Gamma) - P(\Phi), I[\Phi] + \delta O[\Phi] \ast O'[\Phi])$ into a term independent of $\delta$ -- which is clearly $I_{i,k}[\Gamma]$ -- and term weighted by $\delta$. We now focus on the second, interesting term.
The Feynman diagram expansion of this expression consists of graphs with two special vertices, labelled by $O[\Phi]$ and $O'[\Phi]$; and any number of other vertices, labelled by $I[\Phi]$. Each edge is labelled by $P(\Gamma) - P(\Phi)$. Further, every connected component of such a graph must contain either the vertex labelled by $O[\Phi]$, or that labelled by $O'[\Phi]$. Thus, the relevant graphs have only one or two connected components.
The graphs which have precisely two connected components produce $O[\Gamma] \ast O'[\Gamma]$. Thus, to prove the first part of the lemma, it suffices to show that the weights of graphs involving one connected component vanish for $\Phi, \Gamma$ sufficiently small, with $\Phi < \Gamma'$.
Graphs with one connected component must have a chain of edges connecting the two special vertices. The length of any such chain is bounded by an expression involving only $i$ and $k$. Let us choose arbitrarily a metric on $M$. By taking $\Phi$ and $\Gamma$ to be sufficiently small, we can assume that the support of the propagator on each edge is within $\epsilon$ of the diagonal, where, by choosing $\Gamma$ sufficiently small, we can take $\epsilon$ as small as we like. Similarly, the support of the $I_{r,s}[\Gamma]$ labelling a vertex of genus $r$ and valency $s$ can be taken to be within $c_{r,s} \epsilon$ of the diagonal, where $c_{r,s}$ is a cominatorial constant. In addition, by choosing $\Phi$ to be small enough we can ensure that the supports of $O[\Phi]$ and $O'[\Phi]$ is disjoint.
It follows that, by choosing $\Phi$ and $\Gamma$ to be sufficiently small, the weight of any connected graph is obtained by contracting a distribution and a function which have disjoint support, and is thus zero.
Thus we have proved the first part of the lemma, and produced a map
$$
\operatorname{Obs}^{k}(U) \otimes \operatorname{Obs}^{l}(V) \to \operatorname{Obs}^{k+l}(U \amalg V).
$$
It is a simple exercise to show that this is a cochain map, and satisfies the associativity and commutativity properties necessary to define a prefactorization algebra.
In this page we will prove the following result.
''Theorem.'' //For any $n$, the natural map//
$$
\til{\Obs}^{cl}(n)(U) \to \til{\Obs}^{cl}(U)^{\otimes n}
$$
//is a quasi-isomorphism. Further, the natural map//
$$
\til{\Obs}^{cl}(1) (U) \to \Obs^{cl}(U)
$$
//is a quasi-isomorphism.//
''Proof.'' Recall that $\br{\E}_c(U)$ refers to compactly supported distributional sections of the bundle $E$ of fields on $U$. Similarly, $\E_c(U)$ refers to compactly supported smooth sections, and $\E^!_c(U)$ and $\br{\E}^!_c(U)$ refer to the corresponding sections of the bundle $E^! = E \otimes \op{Dens}_U$.
Let $Q : \br{\E}^!_c(U) \to \br{\E}^!_c(U)$ (or, $\E_c^!(U) \to \E_c^!(U)$) refer to the linearized differential. The operator $Q$ comes from a differential operator on the vector bundle $E$, making $E$ into an [[elliptic complex]]. A result of [[Atiyah-Bott]] implies that the inclusion $(\E_c^!,Q) \into (\br{\E}^!_c(U), Q)$ is a homotopy equivalence.
We can decompose $\til{\Obs}^{cl}(U)$ into a product
$$
\til{\Obs}^{cl}(U) = \prod_i \til{\Obs}^{cl}_i(U).
$$
The differential maps $\til{\Obs}^{cl}_i(U)$ to $\prod_{j \ge i} \til{\Obs}^{cl}_j(U)$. The complex $\Obs^{cl}(U)$ has a similar product decomposition, and the map $\til{\Obs}^{cl}(U)\to\Obs^{cl}(U)$ is compatible with this product decomposition.
Thus, it suffices to show that the map
$$
\til{\Obs}^{cl}_i(U) \to \Obs^{cl}_i(U)
$$
is a homotopy equivalence. Here, both sides are equipped with a linearized differential arising from the elliptic operator $Q$.
The complex $\til{\Obs}_i^{cl}(U)$ is defined by a pull-back diagram
$$
\begin{array} { c c c }
\til{\Obs}_i^{cl}(U) & \to & \Obs_i^{cl}(U) \otimes \E^!_c(U) \\
\downarrow & & \downarrow \\
\Obs^{cl}_i(U) & \to & \Obs^{cl}_i(U) \otimes \br{\E}^!_c(U).
\end{array}
$$
Since the map $\E^!_c(U) \to \br{\E}^!_c(U)$ is a homotopy equivalence, it follows easily that the map $\til{\Obs}^{cl}_i(U) \to \Obs^{cl}_i(U)$ is a homotopy equivalence.
This implies that the map
$$
\til{\Obs}^{cl}(U) \to \Obs^{cl}(U)
$$
is a homotopy equivalence. This proves the second statement in the theorem.
Now, it remains to show that the map
$$
\til{\Obs}^{cl}(n)(U) \to \Obs^{cl}(U)^{\otimes n}
$$
is a homotopy equivalence. Since the projective tensor product commutes with limits, it suffices to verify that the map
$$
\til{\Obs}^{cl}_{i}(U) \br{\otimes} \til{\Obs}^{cl}_{j}(U) \to {\Obs}^{cl}_{i}(U) \otimes {\Obs}^{cl}_{j}(U)
$$
is a homotopy equivalence. Again, using the homotopy equivalence between ${\Obs}^{cl}_{i}(U)$ and $\til{\Obs}^{cl}_{i}(U)$, it suffices to verify that the map
$$
{\Obs}^{cl}_{i}(U) \br{\otimes} {\Obs}^{cl}_{j}(U) \to {\Obs}^{cl}_{i}(U) \otimes {\Obs}^{cl}_{j}(U)
$$
is a homotopy equivalence.
Since ${\Obs}^{cl}_{i}(U)$ is the space of $S_i$ invariant compactly supported distributional sections of an elliptic complex on $U^i$, this will follow from a general lemma concerning elliptic complexes.
''Lemma.''// Let $U,V$ be manifolds and let $(F,\d_F)$ and $(G,\d_G)$ be graded bundles on $U$ and $V$ respectively, equipped with differential operators making the spaces of sections into elliptic complexes. $\Gamma_c(U,F)$ and $\br{\Gamma}_c(U,F)$ will denote the complexes of compactly supported smooth and distributional sections, respectively. //
//Then the map//
$$
\br{\Gamma}_c(U,F) \br{\otimes } \br{\Gamma}_c(V,G) \to \br{\Gamma}_c(U,F) {\otimes } \br{\Gamma}_c(V,G)
$$
//is a homotopy equivalence. //
''Proof of lemma.''
Now,
$$
\br{\Gamma}_c(U,F) {\otimes } \br{\Gamma}_c(V,G) = \br{\Gamma}_c( U \times V, F \boxtimes G).
$$
There is a homotopy equivalence
$$
\br{\Gamma}_c(U,F) \simeq \Gamma_c(U,F)
$$
and similarly for $G$. Thus,
$$
\br{\Gamma}_c(U,F) \br{\otimes } \br{\Gamma}_c(V,G) \simeq {\Gamma}_c(U,F) \br{\otimes } {\Gamma}_c(V,G) .
$$
Since the inductive tensor product commutes with colimits,
$$
{\Gamma}_c(U,F) \br{\otimes } {\Gamma}_c(V,G) = \colim_{K,L} {\Gamma}(K,F) \br{\otimes } {\Gamma}_c(L,G)
$$
where the colimit is over compact subsets $K \subset U$, $L \subset V$. The space ${\Gamma}(K,F)$ is a Frechet space. Inductive and projective tensor products coincide on Frechet spaces. Thus,
$$
\begin{split}
{\Gamma}_c(U,F) \br{\otimes } {\Gamma}_c(V,G) &= \colim_{K,L} \left( {\Gamma}(K,F) \otimes {\Gamma}_c(L,G) \right)\\
&=\colim_{K,L} \Gamma (K \times L, F \boxtimes G ) \\
&= \Gamma_c( U \times V, F \boxtimes G).
\end{split}
$$
Finally, since $F \boxtimes G$ is an elliptic complex on $U \times V$, there is a homotopy equivalence
$$
\Gamma_c( U \times V, F \boxtimes G) \simeq \br{\Gamma}_c( U \times V, F \boxtimes G) .
$$
$\square$
This completes the proof of the theorem.
$\square$
!! Pull back
Let $\F$ be a factorization algebra on $M$. Let $U \subset M$ be an open subset. Then we can restrict $\F$ to a factorization algebra $\F\mid_U$ on $U$, whose value on an open subset $V \subset U$ is simply $\F(V)$.
In this page we will discuss a generalization of this construction. We will not try to define pull-backs for arbitrary maps, but only for open immersions.
Let $f : N \to M$ be an open immersion. Let $\mf{U}_f$ be the cover of $N$ consisting of those open subsets $U \subset N$ with the property that
$$
f \mid_U : U \to f(U)
$$
is a homeomorphism. (To say that $f$ is an open immersion means that sets of this form cover $N$).
Now, $\mf{U}_f$ is a factorizing basis for $N$. Let us define a $\mf{U}_f$-prefactorization algebra $f^\ast \F$ by
$$
f^\ast \F(U) = \F( f(U))
$$
if $U \in \mf{U}_f$.
''Lemma.'' $f^\ast \F$ is a $\mf{U}_f$-factorization algebra.
''Proof.'' We need to verify that if $U \in \mf{U}_f$, and $\mf{V}$ is a factorizing cover of $U$ by elements of $\mf{U}_f$, that
$$
\check{C}( \mf{V}, f^\ast \F ) \simeq f^\ast \F(U) = \F( f(U)).
$$
Now, $f (\mf{V})$ is a factorizing cover of $f(U)$, and
$$
\check{C}( \mf{V}, f^\ast \F ) = \check{C}( f(\mf{V}, \F).
$$
The result follows from the fact that $\F$ is a factorization algebra on $M$.
$\square$
So far we have defined $f^\ast \F$ as a $\mf{U}_f$-factorization algebra. We can [[extend | extension from a basis]] $f^\ast \F$ to an actual factorization algebra, which we will continue to call $f^\ast \F$.
A crucial feature of factorization algebras is that they push forward nicely. Let $M$ and $N$ be topological spaces admitting factorizing covers and let $f:M \rightarrow N$ be a continuous map. Given a factorizing cover $\mathfrak U = \{U_\alpha\}$ of an open $U \subset N$, let $f^{-1} \mathfrak U = \{ f^{-1} U_\alpha \}$ denote the preimage cover of $f^{-1} U \subset M$. Observe that $f^{-1} \mathfrak U$ is factorizing: given a finite collection of points $\{ x_1, \ldots, x_n\}$ in $f^{-1} U$, the image points $\{f(x_1), \ldots, f(x_n)\}$ can be covered by a disjoint collection of opens $U_{\alpha_1}, \ldots, U_{\alpha_k}$ in $\mathfrak U$ and hence $f^{-1} U_{\alpha_1}, \ldots, f^{-1} U_{\alpha_k}$ is a disjoint collection of opens in $f^{-1} \mathfrak U$ covering the $x_j$.
''Definition''
Given a factorization algebra $\F$ on a space $M$ and a continuous map $f: M \rightarrow N$, the //pushforward factorization algebra// $f_\ast \F$ on $N$ is defined by
\[
f_\ast \F(U) := \F(f^{-1}(U)).
\]
Note that for the map to a point $f: M \rightarrow pt$, the pushforward factorization algebra $f_\ast \F$ is simply the global sections of $\F$. We also call this the //factorization homology// of $\F$ on $M$. We sometimes denote this $\op{FH}(M, \F)$.
We work throughout with the notion of quantum field theory as articulated in \cite{webbook}. Here we give a quick reminder on the definitions and notations that we borrow from there, but not the proofs or motivation. Before diving into the most general definition, we give the definition of scalar field theory, as the basic idea is most clearly understood in that context. With these definitions in hand, the reader can understand the [[local-effective correspondence]], which indicates how to translate between the usual language of Lagrangians and the notion of quantum field theory given here.
The essential idea behind the definition of QFT used here is //effective field theory//. For an exposition of this idea, see the introduction of \cite{webbook}. Here is a gloss of the idea. Any experiment only involves energies of some finite magnitude; as we build bigger accelerators, lasers, and so on, we are able to explore natural phenomenon at greater energy scales. Our theories are then related to the energy scale we wish to describe. For instance, if we are working at scale $E$, we might only take the path integral over fields of energy $\leq E$. Denote the field theory describing phenomena up to scale $E$ by $T_E$. To have a coherent description of a full-blown theory (that is, going up to scale $\infty$), we need a way to relate $T_E$ to $T_{E'}$ for all energies $E$ and $E'$.
Below we give one way to provide a sensible definition of a coherent family of theories, but the parameter is by //length// rather than //energy//.
!! Scalar theories
For simplicity of exposition, let $M$ be a smooth compact oriented Riemannian manifold. Our fields are simply smooth real-valued functions on $M$, which we denote $C^\infty(M)$. Everything we do is perturbative, so we fix a nondegenerate quadratic action
$$
Q(\phi) = -\langle \phi, (D+m^2) \phi \rangle = -\int_M \phi (D+m^2) \phi \, \operatorname{dvol},
$$
where $D$ is the standard Laplacian (the sign is chosen to have nonnegative eigenvalues) and $m$ is a fixed positive number interpreted as "mass." We might call this the "kinetic energy" of the field $\phi$.
The path integral $\int_{fields} e^{-Q(\phi)} \, d\phi$ can be made rigorous, since it is just a Gaussian integral. But we would like to include "interaction terms" in our action functional, which is not amenable to the methods of analysis. Our approach is to use a Feynman diagram expansion, combined with the idea of an effective field theory.
We assume the reader is familiar with the basic process of Feynman diagram computations (for a refresher see "Generalities on Feynman Graphs" in chapter 2 of \cite{webbook}). Our notation is as follows. Let $P$ denote a propagator in $\operatorname{Sym}^2 C^\infty(M)$. Let $I = \sum_{g,i \geq 0} \hbar^g I_{g,i}$ denote a [[ functional|local action functional]], namely an element of $\Oo^+(C^\infty(M))[[\hbar]]$. The index $i$ denotes the homogeneous degree in $\Ool$ and $g$ denotes the "genus" or "loop number," namely a power of $\hbar$. Given a stable graph $\gamma$, let $w_\gamma(P,I)$ denote the //weight// of the graph for our propagator and local action functional. We denote the total sum
$$
W(P,I) = \sum_{\text{connected stable graphs } \gamma} \frac{\hbar^{-\chi (\gamma)}}{|Aut(\gamma)|} w_\gamma (P,I),
$$
which is the total Feynman diagram expansion for our propagator and action functional.
The propagator for scalar field theory is determined by our kinetic action term. Naively, it is the distributional kernel $(D+m)^{-1} \in \mathcal{D}(M \times M)$. But in our approach to effective field theory, we use the Feynman diagram expansion to relate theories at different length scales using the //renormalization group equation//. Let $K_l$ denote the distributional kernel for the operator $e^{-lD}$. We see that
$$
(D+m)^{-1} = \int_{l = 0}^\infty e^{-m^2l} K_l \, dl.
$$
We define a propagator from length scale $\epsilon$ to $L$ by $P(\epsilon, L) = \int_\epsilon^L e^{-m^2l} K_l \, dl$. If we have a functional $I[\epsilon] \in \Oo(C^\infty(M)$ that describes our "theory at scale $\epsilon$," then the //renormalization group flow// to scale $L$ is the functional
$$
I[L] := W(P(\epsilon,L), I[\epsilon]).
$$
Thus, we have a precise description of a "coherent family of effective actions." With this description in hand, we suggest the following definition.
''Definition''
//A// perturbative quantum field theory //is given by a family of effective interactions $\{I[L]\}_{L \in (0,\infty]}$ such that//
# //the renormalization group equation//
$$
I[L] = W(P(\epsilon,L),I[\epsilon])
$$
//holds for all $\epsilon, L \in (0,\infty]$;//
# //for every $g, i$, there is a small $L$ asymptotic expansion//
$$
I_{g,i}[L] \simeq \sum_{r = 0}^{\infty} g_r(L) \Phi_r
$$
//where $g_r(L)$ is a smooth function of $L \in (0,\infty)$ and $\Phi \in \Ool(C^\infty(M))$.//
We denote the set of perturbative quantum field theories by $\mathcal{T}^{(\infty)}$. Let $\mathcal{T}^{(n)}$ denote the set of theories defined modulo $\hbar^{n+1}$. Hence, $\mathcal{T}^{(\infty)} = \underset{\leftarrow}{\lim} \, \mathcal{T}^{(n)}$.
What does the second condition, "a small $L$ asymptotic expansion," mean? In practice, the action functionals that physicists write down are [[ local|local action functional]]. Hence, our definition of perturbative QFT includes the condition that as the length scale goes to $0$, and all fields are included in the path integral, our effective actions "converge" to a local action functional. Our notion of "convergence" is that we have an asymptotic expansion at $0$ as follows.
''Definition''
//Given a family of homogeneous functionals $I_k[L] \in \operatorname{Sym}^k (C^\infty(M)^\vee)$, parametrized by $L \in (0,\infty)$, a// small $L$ asymptotic expansion //is a non-decreasing sequence of nonnegative integers $\{d_R\}$, tending to infinity, and functions $g_r$ and $\Phi_r$ as above, such that for every $R$ and for every field $\phi \in C^\infty(M)$,//
$$
\lim_{L \rightarrow 0} L^{-d_R} \left( I_k[L](\phi) - \sum_{r=0}^R g_r(L) \Phi_r(\phi)\right) = 0.
$$
Such theories are in bijective correspondence with the local action functionals, a result we call the [[local-effective correspondence]].
!! The general case
In the previous section we tried to give the cleanest description of the essential idea, by focusing on how perturbative and effective field theories combine in the context of scalar field theory. Here we will strive for maximal generality. We want to extend our definition in the following ways:
* allow //noncompact// manifolds, not just compact manifolds;
* allow smooth sections of a $\mathbb{Z}$-graded super //vector bundle// as fields, not just smooth real-valued functions;
* allow //families// of field theories parametrized by a manifold $X$, rather than just a single theory;
* allow //sheaves of theories// rather than a single set $\mathcal{T}$ of theories.
These extensions make the definition more complicated, but hopefully with these aims in mind, it will still be clear. We give a gloss of the necessary ingredients to give the full-blown definition. For a thorough exposition of the definition, we refer the reader to \cite{webbook}, of course.
!!!! Allowing vector bundles
The modification here is quite modest. Let $E$ denote a $\mathbb{Z}$-graded super vector bundle on $M$, a smooth Riemannian manifold (not necessarily compact). We require $E$ to be of finite total rank. Let $\E$ denote the space of smooth sections.
We will eventually want the freedom to decompose $E$ into two pieces, $E_1 \oplus E_2$, whose respective sections form the "propagating" and "nonpropagating" fields. The quadratic part of the action functional -- the "kinetic energy" -- only depends on the propagating fields, and so the propagators will only involve the propagating fields. Hence, when we run the renormalization group equation, we only modify the "propagating part" of a functional. What role does this splitting play? One natural application is that nonpropagating fields allow us to include fixed background fields (e.g., to describe how a charged particle moves in a fixed electromagnetic field).
There are then two parts to defining a perturbative quantum field theory. First, we need to specify a "free" theory whose action functional is purely quadratic. The free theory pins down the propagator that appears in Feynman diagrams and the RG flow. The crucial example for us is a [[free BV theory | BV theory]]. Second, we study theories "perturbing" the free theory, i.e., with interaction terms in their action functionals. Again, the crucial example is a [[BV theory]].
!!!! Allowing families
We would like to study theories in families over a parameter space $X$. In the spirit of algebraic geometry, we would like $X$ to be a manifold, possibly with nilpotent directions that will allow us to study formal deformations. Hence we introduce the following definition.
''Definition''
A nilpotent graded manifold $(X,A)$ is a pair:
# a smooth manifold with corners $X$;
# a sheaf $A$ of commutative differential graded superalgebras over the structure sheaf $C^\infty_X$;
such that
# $A$ is locally free of finite rank as a $C^\infty_X$-module;
# $A$ is equipped with an ideal sheaf $I$ such that $A/I = C^\infty_X$ and $I^k = 0$ for some integer $k > 0$. The sheaf $I$, its powers $I^n$, and the quotient sheaves $A/I^n$ are all locally free $C^\infty_X$-modules.
We denote the smooth global sections of $A$ by $\mathscr{A}$.
All the field-theoretic constructions will be $\mathscr{A}$-linear, which we interpret as describing $(X,A)$-families of field theories. For instance, the role of the Laplacian in scalar field theory is taken by an even, $\mathscr{A}$-linear, order-two differential operator
\[
\operatorname{D}: \E_1 \otimes \mathscr{A} \rightarrow \E_1 \otimes \mathscr{A}
\]
that is a generalized Laplacian.
One interesting consequence of allowing nilpotent directions is that we can include linear and quadratic terms to appear in the interaction part of the action functional, so long as they are weighted by elements of the nilpotent ideal $I$.
!!!! Allowing noncompact manifolds
Parametrizing the effective field theories by length scale handles the //ultraviolet// divergences that appear in the perturbative approach, but infrared divergences appear on a noncompact manifold. A simple idea resolves this issue: modify the propagator to prevent infrared divergences by multiplying the heat kernel against a cut-off function that is supported in a small neighborhood of the diagonal in $M^2$. On a noncompact manifold, there are many different heat kernels, depending on the boundary behavior we require of solutions to the PD, but the small-time asymptotics of the heat kernel at the diagonal are independent of boundary conditions, and this data is what makes the local-effective correspondence possible. In essence, all the earlier work carries over if one uses the cut-off kernel in place of the actual heat kernel for Feynman diagram computations. In \cite{webbook}, it is shown that the space of theories is independent of the choice of cut-off kernel.
Our use of [[parametrices]] in this wiki is essentially the same as their use in constructing field theories on noncompact manifolds.
!!!! Allowing sheaves
Cut-off kernels make it possible to attempt local-to-global constructions of theories. Since the notion of theory is independent of cut-off kernel, one can show that theories restrict from larger open sets to smaller open sets. Using the technique of counterterms, one can then show that the space of theories forms a sheaf over the spacetime manifold. Again, our use of parametrices here to restrict our constructions to an open subset of the manifold is the same as their use in constructing field theories.
Another aim of the wiki is to relate quantum field theory, as developed in \cite{webbook}, to factorization algebras. We give a natural definition of an //observable// of a quantum field theory, which leads to the following theorem.
''Theorem''
//For a [[classical field theory|definition of classical field theory]] and a choice of [[BV quantization|BV theory]], the quantum observables $\Obs^q$ form a factorization algebra over the ring $\R[[\hbar]]$. Moreover, the factorization algebra of classical observables $\Obs^{cl}$ is homotopy equivalent to the quotient $\Obs^q/\hbar$ as a factorization algebra.//
Thus, the quantum observables form a factorization algebra and, in a very weak sense, are related to the classical observables. The quantization theorems will sharpen the relationship between classical and quantum observables.
!! Guide to reading
# For a reminder on what we mean by a quantum field theory, look [[here|overview of perturbative quantum field theory]].
# Parts (c) - (h) of the section on quantum field theory construct the factorization algebra of quantum observables.
Note that this theorem is proved as part of proving the [[the weak quantization theorem]], to which part (h) is devoted.
In the next few pages, we will prove the [[first version | the weak quantization theorem]] of our quantization theorem. This will be achieved by construction, associated to a quantum field theory on $M$ in the sense of \cite{webbook}, a factorization algebra on $M$. This is a quantization (in the weak sense) of the $P_0$ factorization algebra associated to the corresponding classical field theory.
//Remark:// We will denote the factorization algebra of quantum observables simply by $\Obs$. The classical observables are denoted $\Obs^{cl}$.
!!! Outline of the construction
# [[Local quantum observables]] Here, we identify those observables which are supported on some open subset $U \subset M$.
# [[The prefactorization algebra of observables]] This page describes the prefactorization algebra structure on the observables.
# [[The factorization algebra of observables]] Here we verify that the prefactorization algebra of observables is in fact a factorization algebra.
Quillen, Rational homotopy theory, 1969
<<<
''Warning:'' //The proofs of the results stated in this page are not fully written yet. //
<<<
In this page we will state the converse to our quantization theorem. We will see that the quantizations which we construct of the $P_0$ factorization algebras associated to a classical field theory are actually quasi-isomorphic to BD quantizations. We conjecture that every BD quantization arises in this way.
!!! The quantization theorem
Let us suppose we have a classical field theory on a manifold $M$, with associated $P_0$ factorization algebra $\Obs^{cl}$.
''Theorem''
# //Every quantum field theory, in the sense of \cite{webbook}, yields a lax factorization BD algebra $\ObsHomotopy^q$ on $M$, in the category of good modules over $\A[[\hbar]]$. (In this statement, the adjective lax refers to the factorization structure and not to the BD structure). //
#// Modulo $\hbar$, $\ObsHomotopy^q(U)$ is a $P_0$ factorization algebra which is equivalent to the $P_0$ factorization algebra of classical observables. //
''Conjecture'' (Vague statement) //Every BD quantization of $\Obs^{cl}$ arises in this way//.
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// // REPLACE story.search() for option to "show search results in a list"
//{{{
Story.prototype.search = function(text,useCaseSensitive,useRegExp)
{
var co=config.options; // abbrev
var re=new RegExp(useRegExp ? text : text.escapeRegExp(),useCaseSensitive ? "mg" : "img");
if (config.options.chkSearchHighlight) highlightHack=re;
var matches = store.search(re,co.chkSearchByDate?"modified":"title","");
if (co.chkSearchByDate) matches=matches.reverse(); // most recent first
var q = useRegExp ? "/" : "'";
clearMessage();
if (!matches.length) {
if (co.chkSearchListTiddler) discardSearchResults();
displayMessage(config.macros.search.failureMsg.format([q+text+q]));
} else {
if (co.chkSearchList||co.chkSearchListTiddler)
reportSearchResults(text,matches);
else {
var titles = []; for(var t=0; t<matches.length; t++) titles.push(matches[t].title);
this.closeAllTiddlers(); story.displayTiddlers(null,titles);
displayMessage(config.macros.search.successMsg.format([matches.length, q+text+q]));
}
}
highlightHack = null;
}
//}}}
// // REPLACE store.search() for enhanced searching/sorting options
//{{{
TiddlyWiki.prototype.search = function(searchRegExp,sortField,excludeTag,match)
{
var co=config.options; // abbrev
var tids = this.reverseLookup("tags",excludeTag,!!match,sortField);
var opened=[]; story.forEachTiddler(function(tid,elem){opened.push(tid);});
// eliminate tiddlers tagged with excluded tags
if (co.chkSearchExcludeTags&&co.txtSearchExcludeTags.length) {
var ex=co.txtSearchExcludeTags.readBracketedList();
var temp=[]; for(var t=tids.length-1; t>=0; t--)
if (!tids[t].tags.containsAny(ex)) temp.push(tids[t]);
tids=temp;
}
// scan for matching titles first...
var results = [];
if (co.chkSearchTitles) {
for(var t=0; t<tids.length; t++) {
if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
if(tids[t].title.search(searchRegExp)!=-1) results.push(tids[t]);
}
if (co.chkSearchShadows)
for (var t in config.shadowTiddlers) {
if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
if ((t.search(searchRegExp)!=-1) && !store.tiddlerExists(t))
results.push((new Tiddler()).assign(t,config.shadowTiddlers[t]));
}
}
// then scan for matching text, tags, or field data
for(var t=0; t<tids.length; t++) {
if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
if (co.chkSearchText && tids[t].text.search(searchRegExp)!=-1)
results.pushUnique(tids[t]);
if (co.chkSearchTags && tids[t].tags.join(" ").search(searchRegExp)!=-1)
results.pushUnique(tids[t]);
if (co.chkSearchFields && store.forEachField!=undefined)
store.forEachField(tids[t],
function(tid,field,val) {
if (val.search(searchRegExp)!=-1) results.pushUnique(tids[t]);
},
true); // extended fields only
}
// then check for matching text in shadows
if (co.chkSearchShadows)
for (var t in config.shadowTiddlers) {
if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
if ((config.shadowTiddlers[t].search(searchRegExp)!=-1) && !store.tiddlerExists(t))
results.pushUnique((new Tiddler()).assign(t,config.shadowTiddlers[t]));
}
// if not 'titles first', or sorting by modification date,
// re-sort results to so titles, text, tag and field matches are mixed together
if(!sortField) sortField = "title";
var bySortField=function(a,b){
if(a[sortField]==b[sortField])return(0);else return(a[sortField]<b[sortField])?-1:+1;
}
if (!co.chkSearchTitlesFirst || co.chkSearchByDate) results.sort(bySortField);
return results;
}
//}}}
// // HIJACK core {{{<<search>>}}} macro to add "report" and "simple inline" output
//{{{
config.macros.search.SOP_handler=config.macros.search.handler;
config.macros.search.handler = function(place,macroName,params)
{
// if "report", use SearchOptionsPlugin report generator for inline output
if (params[1]&¶ms[1].substr(0,6)=="report") {
var keyword=params[0];
var options=params[1].split("=")[1]; // split "report=option+option+..."
var heading=params[2]?params[2].unescapeLineBreaks():"";
var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
if (matches.length) wikify(heading+window.formatSearchResults(keyword,matches,options),place);
} else if (params[1]) {
var keyword=params[0];
var heading=params[1]?params[1].unescapeLineBreaks():"";
var seperator=params[2]?params[2].unescapeLineBreaks():", ";
var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
if (matches.length) {
var out=[];
for (var m=0; m<matches.length; m++) out.push("[["+matches[m].title+"]]");
wikify(heading+out.join(seperator),place);
}
} else
config.macros.search.SOP_handler.apply(this,arguments);
};
//}}}
// // SearchResults panel handling
//{{{
setStylesheet(".searchResults { padding:1em 1em 0 1em; }","searchResults"); // matches std tiddler padding
config.macros.search.createPanel=function(text,matches,body) {
function getByClass(e,c) { var d=e.getElementsByTagName("div");
for (var i=0;i<d.length;i++) if (hasClass(d[i],c)) return d[i]; }
var panel=createTiddlyElement(null,"div","searchPanel","searchPanel");
this.renderPanel(panel,text,matches,body);
var oldpanel=document.getElementById("searchPanel");
if (!oldpanel) { // insert new panel just above tiddlers
var da=document.getElementById("displayArea");
da.insertBefore(panel,da.firstChild);
} else { // if panel exists
var oldwrap=getByClass(oldpanel,"searchResults");
var newwrap=getByClass(panel,"searchResults");
// if no prior content, just insert new content
if (!oldwrap) oldpanel.insertBefore(newwrap,null);
else { // swap search results content but leave containing panel intact
oldwrap.style.display='block'; // unfold wrapper if needed
var i=oldwrap.getElementsByTagName("input")[0]; // get input field
if (i) { var pos=this.getCursorPos(i); i.onblur=null; } // get cursor pos, ignore blur
oldpanel.replaceChild(newwrap,oldwrap);
panel=oldpanel; // use existing panel
}
}
this.showPanel(true,pos);
return panel;
}
config.macros.search.renderPanel=function(panel,text,matches,body) {
var wrap=createTiddlyElement(panel,"div",null,"searchResults");
wrap.onmouseover = function(e){ addClass(this,"selected"); }
wrap.onmouseout = function(e){ removeClass(this,"selected"); }
// create toolbar: "open all", "fold/unfold", "close"
var tb=createTiddlyElement(wrap,"div",null,"toolbar");
var b=createTiddlyButton(tb, "open all", "open all matching tiddlers", function() {
story.displayTiddlers(null,this.getAttribute("list").readBracketedList()); return false; },"button");
var list=""; for(var t=0;t<matches.length;t++) list+='[['+matches[t].title+']] ';
b.setAttribute("list",list);
var b=createTiddlyButton(tb, "fold", "toggle display of search results", function() {
config.macros.search.foldPanel(this); return false; },"button");
var b=createTiddlyButton(tb, "close", "dismiss search results", function() {
config.macros.search.showPanel(false); return false; },"button");
createTiddlyText(createTiddlyElement(wrap,"div",null,"title"),"Search for: "+text); // title
wikify(body,createTiddlyElement(wrap,"div",null,"viewer")); // report
return panel;
}
config.macros.search.showPanel=function(show,pos) {
var panel=document.getElementById("searchPanel");
var i=panel.getElementsByTagName("input")[0];
i.onfocus=show?function(){config.macros.search.stayFocused(true);}:null;
i.onblur=show?function(){config.macros.search.stayFocused(false);}:null;
if (show && panel.style.display=="block") { // if shown, grab focus, restore cursor
if (i&&this.stayFocused()) { i.focus(); this.setCursorPos(i,pos); }
return;
}
if(!config.options.chkAnimate) {
panel.style.display=show?"block":"none";
if (!show) { removeChildren(panel); config.macros.search.stayFocused(false); }
} else {
var s=new Slider(panel,show,false,show?"none":"children");
s.callback=function(e,p){e.style.overflow="visible";}
anim.startAnimating(s);
}
return panel;
}
config.macros.search.foldPanel=function(button) {
var d=document.getElementById("searchPanel").getElementsByTagName("div");
for (var i=0;i<d.length;i++) if (hasClass(d[i],"viewer")) var v=d[i]; if (!v) return;
var show=v.style.display=="none";
if(!config.options.chkAnimate)
v.style.display=show?"block":"none";
else {
var s=new Slider(v,show,false,"none");
s.callback=function(e,p){e.style.overflow="visible";}
anim.startAnimating(s);
}
button.innerHTML=show?"fold":"unfold";
return false;
}
config.macros.search.stayFocused=function(keep) { // TRUE/FALSE=set value, no args=get value
if (keep===undefined) return this.keepReportInFocus;
this.keepReportInFocus=keep;
return keep
}
config.macros.search.getCursorPos=function(i) {
var s=0; var e=0; if (!i) return { start:s, end:e };
try {
if (i.setSelectionRange) // FF
{ s=i.selectionStart; e=i.selectionEnd; }
if (document.selection && document.selection.createRange) { // IE
var r=document.selection.createRange().duplicate();
var len=r.text.length; s=0-r.moveStart('character',-100000); e=s+len;
}
}catch(e){};
return { start:s, end:e };
}
config.macros.search.setCursorPos=function(i,pos) {
if (!i||!pos) return; var s=pos.start; var e=pos.end;
if (i.setSelectionRange) //FF
i.setSelectionRange(s,e);
if (i.createTextRange) // IE
{ var r=i.createTextRange(); r.collapse(true); r.moveStart("character",s); r.select(); }
}
//}}}
// // SearchResults report generation
// note: these functions are defined globally, so they can be more easily redefined to customize report formats//
//{{{
if (!window.reportSearchResults) window.reportSearchResults=function(text,matches)
{
var cms=config.macros.search; // abbrev
var body=window.formatSearchResults(text,matches);
if (!config.options.chkSearchListTiddler) // show #searchResults panel
window.scrollTo(0,ensureVisible(cms.createPanel(text,matches,body)));
else { // write [[SearchResults]] tiddler
var title=cms.reportTitle;
var who=config.options.txtUserName;
var when=new Date();
var tags="excludeLists excludeSearch temporary";
var tid=store.getTiddler(title); if (!tid) tid=new Tiddler();
tid.set(title,body,who,when,tags);
store.addTiddler(tid);
story.closeTiddler(title);
story.displayTiddler(null,title);
}
}
if (!window.formatSearchResults) window.formatSearchResults=function(text,matches,opt)
{
var body='';
var title=config.macros.search.reportTitle
var q = config.options.chkRegExpSearch ? "/" : "'";
if (!opt) var opt="all";
var parts=opt.split("+");
for (var i=0; i<parts.length; i++) { var p=parts[i].toLowerCase();
if (p=="again"||p=="all") body+=window.formatSearchResults_again(text,matches);
if (p=="summary"||p=="all") body+=window.formatSearchResults_summary(text,matches);
if (p=="list"||p=="all") body+=window.formatSearchResults_list(text,matches);
if (p=="buttons"||p=="all") body+=window.formatSearchResults_buttons(text,matches);
}
return body;
}
if (!window.formatSearchResults_again) window.formatSearchResults_again=function(text,matches)
{
var title=config.macros.search.reportTitle
var body='';
// search again
body+='{{span{<<search "'+text.replace(/"/g,'"')+'">> /%\n';
body+='%/<html><input type="button" value="search again"';
body+=' onclick="var t=this.parentNode.parentNode.getElementsByTagName(\'input\')[0];';
body+=' config.macros.search.doSearch(t); return false;">';
if (!config.options.chkSearchResultsOptions) { // omit "options..."
body+='</html>}}}\n\n';
return body;
}
body+=' <a href="javascript:;" onclick="';
body+=' var e=this.parentNode.nextSibling;';
body+=' var show=e.style.display!=\'block\';';
body+=' if(!config.options.chkAnimate) e.style.display=show?\'block\':\'none\';';
body+=' else anim.startAnimating(new Slider(e,show,false,\'none\'));';
body+=' return false;">options...</a>';
body+='</html>@@display:none;border-left:1px dotted;margin-left:1em;padding:0;padding-left:.5em;font-size:90%;/%\n';
body+=' %/<<option chkSearchTitles>>titles /%\n';
body+=' %/<<option chkSearchText>>text /%\n';
body+=' %/<<option chkSearchTags>>tags /%\n';
body+=' %/<<option chkSearchFields>>fields /%\n';
body+=' %/<<option chkSearchShadows>>shadows\n';
body+=' <<option chkCaseSensitiveSearch>>case-sensitive /%\n';
body+=' %/<<option chkRegExpSearch>>text patterns /%\n';
body+=' %/<<option chkSearchByDate>>sorted by date\n';
body+=' <<option chkSearchHighlight>> highlight matching text in displayed tiddlers\n';
body+=' <<option chkIncrementalSearch>>incremental key-by-key search: /%\n';
body+=' %/{{twochar{<<option txtIncrementalSearchMin>>}}} or more characters, /%\n';
body+=' %/{{threechar{<<option txtIncrementalSearchDelay>>}}} msec delay\n';
body+=' <<option chkSearchOpenTiddlers>> search only in tiddlers that are currently displayed\n';
body+=' <<option chkSearchExcludeTags>>exclude tiddlers tagged with:\n';
body+=' {{editor{<<option txtSearchExcludeTags>>}}}/%\n';
body+='%/@@}}}\n\n';
return body;
}
if (!window.formatSearchResults_summary) window.formatSearchResults_summary=function(text,matches)
{
// summary: nn tiddlers found matching '...', options used
var body='';
var co=config.options; // abbrev
var title=config.macros.search.reportTitle
var q = co.chkRegExpSearch ? "/" : "'";
body+="''"+config.macros.search.successMsg.format([matches.length,q+"{{{"+text+"}}}"+q])+"''\n";
var opts=[];
if (co.chkSearchTitles) opts.push("titles");
if (co.chkSearchText) opts.push("text");
if (co.chkSearchTags) opts.push("tags");
if (co.chkSearchFields) opts.push("fields");
if (co.chkSearchShadows) opts.push("shadows");
if (co.chkSearchOpenTiddlers) body+="^^//search limited to displayed tiddlers only//^^\n";
body+="~~ searched in "+opts.join(" + ")+"~~\n";
body+=(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"^^ using ":"")
+(co.chkCaseSensitiveSearch?"case-sensitive ":"")
+(co.chkRegExpSearch?"pattern ":"")
+(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"matching^^\n":"");
return body;
}
if (!window.formatSearchResults_list) window.formatSearchResults_list=function(text,matches)
{
// bullet list of links to matching tiddlers
var body='';
var co=config.options; // abbrev
var pattern=co.chkRegExpSearch?text:text.escapeRegExp();
var sensitive=co.chkCaseSensitiveSearch?"mg":"img";
var link='{{tiddlyLinkExisting{<html><nowiki><a href="javascript:;" onclick="'
+'if(config.options.chkSearchHighlight)'
+' highlightHack=new RegExp(\x27'+pattern+'\x27.escapeRegExp(),\x27'+sensitive+'\x27);'
+'story.displayTiddler(null,\x27%0\x27);'
+'highlightHack = null; return false;'
+'" title="%2">%1</a></html>}}}';
for(var t=0;t<matches.length;t++) {
body+="* ";
if (co.chkSearchByDate)
body+=matches[t].modified.formatString('YYYY.0MM.0DD 0hh:0mm')+" ";
var title=matches[t].title;
var fixup=title.replace(/'/g,"\\x27").replace(/"/g,"\\x22");
var tid=store.getTiddler(title);
var tip=tid?tid.getSubtitle():''; tip=tip.replace(/"/g,""");
body+=link.format([fixup,title,tip])+'\n';
}
return body;
}
if (!window.formatSearchResults_buttons) window.formatSearchResults_buttons=function(text,matches)
{
// embed buttons only if writing SearchResults to tiddler
if (!config.options.chkSearchListTiddler) return "";
// "open all" button
var title=config.macros.search.reportTitle;
var body="";
body+="@@diplay:block;<html><input type=\"button\" href=\"javascript:;\" "
+"onclick=\"story.displayTiddlers(null,[";
for(var t=0;t<matches.length;t++)
body+="'"+matches[t].title.replace(/\'/mg,"\\'")+"'"+((t<matches.length-1)?", ":"");
body+="],1);\" accesskey=\"O\" value=\"open all matching tiddlers\"></html> ";
// "discard SearchResults" button
body+="<html><input type=\"button\" href=\"javascript:;\" "
+"onclick=\"discardSearchResults()\" value=\"discard "+title+"\"></html>";
body+="@@\n";
return body;
}
if (!window.discardSearchResults) window.discardSearchResults=function()
{
// remove the tiddler
story.closeTiddler(config.macros.search.reportTitle);
store.deleteTiddler(config.macros.search.reportTitle);
store.notify(config.macros.search.reportTitle,true);
}
//}}}
<<search>><<closeAll>><<permaview>> <<newTiddler>><<newJournal "DD MMM YYYY" "journal">><<saveChanges>><<tiddler TspotSidebar>><<slider chkSliderOptionsPanel OptionsPanel "options »" "Change TiddlyWiki advanced options">>
<<tabs txtMainTab "Timeline" "Timeline" TabTimeline "All" "All tiddlers" TabAll "Tags" "All tags" TabTags "More" "More lists" TabMore>>
We use the Koszul rule of signs throughout.
We work with cochain complexes throughout (though we may occasionally write chain complex, we mean a cochain complex). Thus a differential on a graded module always raise the cohomological degree by +1.
Given a graded module $C^\ast$, the shifted module $C[1]$ is obtained from $C$ by pushing everything down a degree. That is,
\[
C[1]^k = C^{k+1}.
\]
Kevin Costello and Owen Gwilliam
Factorization algebras in perturbative quantum field theory
[Sketch]
The aim here is to sketch how we can replace the construction of the BD factorization algebra which uses non-Hausdorff spaces with a better construction, where we construct a BD algebra in some class of lax factorization algebras.
Outline of the construction:
1) First define an inverse system, labelled by $(I,K)$. Consider the finite number of things less than $(I,K)$ by saying that $(i,k) < (I,K)$ if $2i-2 + k < 2I -2 + K$. This means that $(i,k)$ graphs can contribute to the $(I,K)$ effective action. This is under the assumption that we are dealing with a classical theory where the interaction is at least cubic. Can also deal with quadratic terms, as long as they are accompanied by nilpotent elements of the base ring; leads to a slightly more complicated induction, left to the reader.
Let
$$d_{i,k}= 2i-2+k.$$
Note that $d_{i,k}$ is greater than the diameter of the largest trivalent graph of type $(i,k)$. You can see this by taking such a graph, cutting $i$ edges to get a tree with $2i + k$ vertices. The longest such trivalent tree (just vertices arranged in a straight line) has diameter $2i+k - 3$.
2) For a compact subsets $K$ of $U$, let $\F^{i,k}_{K,\delta}$ be the subset of $\Oo^{i,k}(\E) \otimes \Omega^\ast( (0,\delta))$ consisting of those elements which, at $t \in (0,1)$, are supported within $d_{i,k} t$ of $K^i$.
Also have
$$
\F^{\le \chi }_{K,\delta} = \prod_{2i - 2 + k \le \chi} \F^{i,k}_{K,\delta}.
$$
The differential on this object is well-defined.
In general, we have maps between this guys if $K \subset K'$.
3) If
$$\{K_i\} = \{K_1,\ldots, K_n\}$$ is a tuple of disjoint compact sets, let
$$
\F^{i,k}_{\{K_i\}} \subset \F^{i,k}_{\amalg K_i}
$$
be the subsets where the BV operator doesn't have room to cross from one component to the other.
There are some bounds that define this; it doesn't really matter what they are. However, they decrease when we increase the compact sets $K_i$.
The inclusion
$$
\F^{i,k}_{\{K_j\}} \to \F^{i,k}_{\amalg K_j}
$$
is a quasi-isomorphism.
This means that same holds for the map
$$
\F^{\le \chi}_{\{K_j\}} \to \F^{\le \chi}_{\amalg K_i}.
$$
These spaces are functorial with respect to inclusions of compact sets.
Also, there are maps
$$
\F^{\le \chi}_{ \{K_i\}} \otimes \F^{\le \chi}_{ \{K_j\}} \to \F^{\le \chi}_{ \{K_i\} \amalg \{K_j\} }
$$
(as all $\delta$'s appearing on the right hand side are smaller, when we impose the condition on the differential).
3) Now let $\alpha$ indicate a tuple of disjoint open sets. For $K \subset U_\alpha$, let $\{K_\alpha\}$ denote the corresponding tuple of disjoint compact sets.
We define the space of admissible observables to be
$$
\op{hocolim}_{K \subset U_\alpha} \F^{\le \chi}_{\{K\alpha\}}.
$$
Call this $\Obs_h^{\chi} ( U_\alpha)$.
5) If $\alpha = \alpha_1 \amalg \alpha_2$, then there's a product map
$$
\Obs_h ( U_{\alpha _1} ) \otimes \Obs_h( U_{\alpha_2} ) \to \Obs_h(U _\alpha).
$$
We only need define this at the level of simplicial sets, because there's a map from the realization of the tensor product to the tensor product of the realization.
Thus, we only need to define a natural transformation of diagrams, as both sides are certain colimits.
But this we did already.
6) We say that $\beta$ is a //corefinement// of $\alpha$ if $U_\alpha = U_\beta$ and, for each $i \in \alpha$, there is some $j \in \beta$ with $U_i \subset U_j$.
A corefinement induces a quasi-isomorphism
$$
\Obs_h( U_\beta) \to \Obs_h ( U_\alpha).
$$
As, it does so on each compact set appearing in the diagram; the point being that if we have a corefinement like this, the condition on the differentials about separating is strong on the right hand side.
7) A //strict map// $\alpha \to \beta$ is an inclusion $U_\alpha \subset U_\beta$ such that, for each $i \in \alpha$, there is a unique $j \in \beta$ such that $U_i \subset U_j$.
A strict map induces a map
$$
\Obs^{\chi}(U_\alpha) \to \Obs^{\chi}(U_\beta).
$$
8) This should be enough data to construct a lax functor. As, every map (in the multicategory $\op{Fact}_X$) is a composition of the inverse of corefinements, and strict maps.
There are some relations to check, as follows. If $f : \alpha \to \beta$ is a strict map, and $\beta ' \to \beta$ is a corefinement, there is a unique corefinement $\alpha' \to \alpha$ together with a strict map $\alpha' \to \beta'$.
We require that the square relating these 4 maps must commute.
This seems to be generators and relations for the multicategory: strict maps, inverse of corefinements, such that this diagram commutes.
So, to give a functor to the appropriate homotopy category, we just need to assign to a corefinement a quasi-iso. and verify that the above diagram commutes.
All seems to work nicely...
9) The last thing to check is that we find something quasi-iso. to observables as defined in the other way.
To do this, we need to verify that $\F^{i,k}_{K,\delta}$ is quasi-isomorphic to the space of functionals of degree $i$ supported on $K$ (where we just consider the linear part of the differential).
Now we can forget everything except the de Rham differential. Indeed, we have a filtration (at level $(i,k)$) on the space, given by cohomological degree coming from $(E^\vee)^{\otimes k}$. Since this is a finite complex (i.e. bounded above and below), the spectral sequence converges, so we can take the associated graded.
Then we have forgotten the differential coming from $\E$, and we can also (by working locally) assume it is a trivial vector bundle.
The question then becomes: consider the subcomplex of $\cinfty(M) \otimes \Omega^\ast ( (0,1)) $ which is supported on the open set
$$
\{ (x,t) \mid x \text{ within } t \text{ of } K \}.
$$
What is the cohomology (with respect to the de Rham differential) of this?
The answer should just be functions supported on $K$.
There are two things to show: firstly, that $H^0$ is the correct thing, and second that there is no $H^1$.
For $H^0$, any function that is closed for the de Rham differential is constant as a function of $t$, so we're done.
For $H^1$: if we replace $(0,1)$ by $[\eps,1)$ it's easy, because then integrating from $\eps$ to $1$ provides a contracting homotopy which doesn't increase support.
Or, we can consider the long exact sequence in cohomology. The third term consists of germs of functions on the closed set
$$
\{(x,t) \mid d(x,K) \ge t \}
$$
of $M \times (0,1)$ (also, of course, with $1$-forms in the $t$ direction). For the long exact sequence, we first need to consider just $H^0$ of this. Now, $H^0$ here consists germs of functions $f(x,t)$ which are independent of $t$. All such come from an actual function on $M \times (0,1)$, independent of $t$; so we're done!
Just to summarize the last step: we verify that, when we consider our complex $\F^{i,k}_{K,\delta}$, it's cohomology just comes from stuff actually supported on $K$. Thus, there's a spectral sequence converging to $\Obs_h(U)$ whose first term involves a product over $(i,k)$ of homotopy colimits of things supported in $K$. There's a similar spectral sequence for strict observables, except that the first term involves the actual colimit, because it involves functionals supported on $U$, which is a colimit of functionals supported on compact subset $K$.
The actual colimit and homotopy colimit are the same, so we're done.
/***
|''Name:''|SlideShowPlugin|
|''Description:''|Creates a slide show from any number of tiddlers|
|''Author:''|Paulo Soares|
|''Contributors:''|John P. Rouillard|
|''Version:''|2.2.6|
|''Date:''|2010-11-17|
|''Source:''|http://www.math.ist.utl.pt/~psoares/addons.html|
|''Documentation:''|[[SlideShowPlugin Documentation|SlideShowPluginDoc]]|
|''License:''|[[Creative Commons Attribution-Share Alike 3.0 License|http://creativecommons.org/licenses/by-sa/3.0/]]|
|''~CoreVersion:''|2.5.0|
***/
//{{{
if(!version.extensions.SlideShowPlugin) { //# ensure that the plugin is only installed once
version.extensions.SlideShowPlugin = {installed: true};
(function($) {
config.macros.slideShow = {maxTOCLength: 30, separator:'-s-'};
config.formatters.push( {
name: "SlideSeparator",
match: "^"+config.macros.slideShow.separator+"+$\\n?",
handler: function(w) {
createTiddlyElement(w.output,"hr",null,'slideSeparator');
}
});
config.macros.slideShow.text = {
label: "slide show", tooltip: "Start slide show",
quit: {label: "x", tooltip: "Quit the slide show"},
firstSlide: {label: "<<", tooltip: "Go to first slide"},
previous: {label: "<", tooltip: "Go back"},
next: {label: ">", tooltip: "Advance"},
lastSlide: {label: ">>", tooltip: "Go to last slide"},
goto: {label: "Go to slide:"},
resetClock: {tooltip: "Reset the clock"},
overlay: "overlay"
};
config.macros.slideShow.handler = function(place,macroName,params,wikifier,paramString){
var args = paramString.parseParams(null,null,false);
this.label = getParam(args,"label",this.text.label);
this.tooltip = getParam(args,"tooltip",this.text.tooltip);
var onclick = function(){config.macros.slideShow.onClick(place,paramString); return false;};
createTiddlyButton(place,this.label,this.tooltip,onclick);
return false;
}
config.macros.slideShow.onClick = function(place,paramString) {
var slide, cm = config.macros.slideShow;
var title = story.findContainingTiddler(place);
title = title ? title.getAttribute("tiddler") : null;
var args = paramString.parseParams(null,null,false);
title = getParam(args,"tiddler",title);
var argsArray = paramString.readMacroParams();
this.single = ($.inArray('single',argsArray) > -1);
this.clicks = ($.inArray('noClicks',argsArray) < 0);
this.keyboard = ($.inArray('noKeyboard',argsArray) < 0);
this.showAll = ($.inArray('showAll',argsArray) > -1);
this.cycle = ($.inArray('cycle',argsArray) > -1);
this.overlays = ($.inArray('noOverlays',argsArray) < 0);
this.theme = getParam(args,"theme");
this.tag = getParam(args,"tag");
this.toc = getParam(args,"toc","headers");
this.sort = getParam(args,"sort");
this.clockFormat = getParam(args,"clockFormat",'0hh:0mm:0ss');
this.auto = getParam(args,"auto",0);
this.header = getParam(args,"header",title);
this.footer = getParam(args,"footer","");
this.clock = getParam(args,"clock");
this.blocked = 0;
if(this.clock){
var startTime = new Date(0);
this.clockCorrection=startTime.getTimezoneOffset()*60000;
startTime = new Date();
this.clockMultiplier = 1;
this.clockInterval = 0;
var clockType= parseFloat(this.clock);
if(clockType < 0) {
this.clockMultiplier = -1;
this.clockInterval = -clockType*60000;
} else if(clockType == 0){
this.clockCorrection = 0;
startTime = new Date(0);
}
this.clockStartTime=startTime.getTime();
}
this.slides = [];
this.openTiddlers = [];
$("#tiddlerDisplay > *").each(function(){cm.openTiddlers.push($(this).attr('tiddler'))});
var count = 0;
this.slideTOC=[];
if(this.single){
if(!store.tiddlerExists(title)) return;
var newTiddler;
var content = store.getTiddlerText(title).split(cm.separator);
$.each(content, function(){
count++;
newTiddler = new Tiddler();
newTiddler.title ="TempSlide" + count;
newTiddler.tags[0] = "excludeLists";
newTiddler.text = $.trim(this);
newTiddler.fields['doNotSave']= true;
store.addTiddler(newTiddler);
cm.buildTOC(count,newTiddler.title);
cm.slides.push(newTiddler.title);
});
} else {
if(this.tag){
var content = store.getTaggedTiddlers(this.tag,this.sort);
$.each(content, function(){
count++;
cm.buildTOC(count,this.title);
cm.slides.push(this.title);
});
} else {
story.displayTiddler(null,title);
var list = $('[tiddler='+title+']').find('.viewer').find('.tiddlyLinkExisting');
$.each(list,function(){
if(!$(this).parents().hasClass("exclude")){
slide = $(this).attr('tiddlylink');
count++;
cm.buildTOC(count,slide);
cm.slides.push(slide);
}
});
}
}
this.nSlides = this.slides.length;
if(this.nSlides==0) return false;
clearMessage();
this.toggleSlideStyles();
if(!this.showAll){
//Attach the key and mouse listeners
if(this.keyboard && !$("#tiddlerDisplay").hasClass("noKeyboard")) $(document).keyup(cm.keys);
if(this.clicks){
$(document).mouseup(cm.clicker);
document.oncontextmenu = function(){return false;}
}
if(this.clock) this.slideClock=setInterval(this.setClock, 1000);
if(this.auto>0){
this.autoAdvance=setInterval(cm.next, this.auto*1000);
}
this.showSlide(1);
} else {
story.closeAllTiddlers();
story.displayTiddlers(null,this.slides);
$(".tiddler").attr("ondblclick",null);
$(document).keyup(cm.endSlideShow);
}
return false;
}
config.macros.slideShow.buildNavigator = function() {
//Create the navigation bar
var i, slidefooter = $("#controlBar")[0];
if(!slidefooter) return;
$(slidefooter).addClass("slideFooterOff noClicks");
var navigator = createTiddlyElement(slidefooter,"SPAN","navigator");
var buttonBar = createTiddlyElement(navigator,"SPAN","buttonBar");
//Make it so that when the footer is hovered over the class will change to make it visible
$(slidefooter).bind("mouseenter mouseleave", function(e){$(this).toggleClass("slideFooterOff");});
//Create the control buttons for the navigation
createTiddlyButton(buttonBar,this.text.firstSlide.label,this.text.firstSlide.tooltip,this.firstSlide,"button");
createTiddlyButton(buttonBar,this.text.previous.label,this.text.previous.tooltip,this.previous,"button");
createTiddlyButton(buttonBar,this.text.quit.label,this.text.quit.tooltip,this.endSlideShow,"button");
createTiddlyButton(buttonBar,this.text.next.label,this.text.next.tooltip,this.next,"button");
createTiddlyButton(buttonBar,this.text.lastSlide.label,this.text.lastSlide.tooltip,this.lastSlide,"button");
if(this.clock){
if(this.clock == 0){
createTiddlyElement(navigator,"SPAN","slideClock");
} else {
createTiddlyButton(navigator," ",this.text.resetClock.tooltip,this.resetClock,"button","slideClock");
}
this.setClock();
}
var index = createTiddlyElement(slidefooter,"SPAN","slideCounter");
index.onclick = this.toggleTOC;
var toc = createTiddlyElement(slidefooter,"SPAN","toc");
var tocLine;
for(i=0; i<this.slideTOC.length; i++){
$(toc).append(this.slideTOC[i][2]);
tocLine = $(toc.lastChild);
tocLine.addClass("tocLevel"+this.slideTOC[i][1]).css("cursor", "pointer").hover(function () {
$(this).addClass("highlight");}, function () {
$(this).removeClass("highlight");});
tocLine.attr("slide",this.slideTOC[i][0]);
tocLine.click(config.macros.slideShow.showSlideFromTOC);
}
//Input box to jump to specific slide
var tocItem = createTiddlyElement(toc,"DIV","jumpItem",null,this.text.goto.label);
var tocJumpInput = createTiddlyElement(tocItem,"INPUT","jumpInput");
tocJumpInput.type="text";
$(tocJumpInput).keyup(config.macros.slideShow.jumpToSlide);
}
//Used to shorten the TOC fields
config.macros.slideShow.abbreviate = function(label){
if(label.length>this.maxTOCLength) {
var temp = new Array();
temp = label.split(' ');
label = temp[0];
for(var j=1; j<temp.length; j++){
if((label.length+temp[j].length)<=this.maxTOCLength){
label += " " + temp[j];
} else {
label += " ...";
break;
}
}
}
return label;
}
config.macros.slideShow.buildTOC = function(count,title) {
var level = 1, text;
switch(this.toc){
case "headers":
var frag = wikifyStatic(store.getTiddlerText(title));
text = frag.replace(/<div class="comment">.*<\/div>/mg,"");
var matches = text.match(/<h[123456]>.*?<\/h[123456]>/mgi);
if(matches){
for (var j=0; j<matches.length; j++){
level = matches[j].charAt(2);
text = matches[j].replace(/<\/?h[123456]>/gi,"");
text = this.abbreviate(text);
this.slideTOC.push([count,level,"<div>"+text+"</div>"]);
}
}
break;
case "titles":
text = this.abbreviate(title);
this.slideTOC.push([count,level,"<div>"+text+"</div>"]);
}
}
config.macros.slideShow.showSlideFromTOC = function(e) {
var cm = config.macros.slideShow;
var slide = parseInt(e.target.getAttribute('slide'));
$("#toc").hide();
cm.showSlide(slide);
return false;
}
config.macros.slideShow.toggleTOC = function(){
$("#toc").toggle();
$("#jumpInput").focus().val('');
return false;
}
config.macros.slideShow.isInteger = function(s){
for (var i = 0; i < s.length; i++){
// Check that current character is number
var c = s.charAt(i);
if (((c < "0") || (c > "9"))) return false;
}
// All characters are numbers
return true;
}
config.macros.slideShow.jumpToSlide = function(e){
var cm = config.macros.slideShow;
if(e.which==13){
var input= $("#jumpInput").val();
if(cm.isInteger(input) && input>0 && input<=cm.nSlides){
$("#toc").hide();
cm.showSlide(input);
} else {$("#jumpInput").val('');}
}
return false;
}
config.macros.slideShow.toggleSlideStyles = function(){
var contentWrapper = $('#contentWrapper');
if(contentWrapper.hasClass("slideShowMode")){
refreshPageTemplate();
removeStyleSheet("SlideShowStyleSheet");
if(this.theme) removeStyleSheet(this.theme);
} else {
$("#displayArea").prepend('<div id="slideBlanker" style="display:none"></div><div id="slideHeader">'+this.header+'</div><div id="slideFooter">'+this.footer+'</div><div id="controlBar"></div>');
setStylesheet(store.getRecursiveTiddlerText("SlideShowStyleSheet"),"SlideShowStyleSheet");
if(this.theme && store.tiddlerExists(this.theme)){setStylesheet(store.getRecursiveTiddlerText(this.theme),this.theme);}
this.buildNavigator();
}
contentWrapper.toggleClass("slideShowMode");
return false;
}
config.macros.slideShow.showSlide = function(n){
if(this.cycle) {
if(n>this.nSlides) {
n = 1;
} else if(n<1) {
n = this.nSlides;
}
} else {
if(n>this.nSlides || n<1) return;
}
story.closeAllTiddlers();
if(this.clock=='-'){this.resetClock();}
story.displayTiddler(null,String(this.slides[n-1]));
$(".tiddler").attr("ondblclick",null);
$("body").removeClass("slide"+this.curSlide);
this.curSlide = n;
$("body").addClass("slide"+this.curSlide);
$("#slideCounter").text(this.curSlide+"/"+this.nSlides);
if(this.overlays){
var contents = $(".viewer *");
this.numOverlays = 1;
while(1){
if(contents.hasClass(this.text.overlay+this.numOverlays)){
this.numOverlays++;
} else {break;}
}
this.numOverlays--;
this.showOverlay(0);
}
return false;
}
config.macros.slideShow.showOverlay = function(n){
var i, set, cm = config.macros.slideShow;
if(!cm.overlays || cm.numOverlays == 0 || n<0 || n>cm.numOverlays){return;}
for(i=1; i<n; i++){
set = $(".viewer "+"."+cm.text.overlay+i);
set.removeClass("currentOverlay nextOverlay");
set.addClass("previousOverlay");
}
set = $(".viewer "+"."+cm.text.overlay+n);
set.removeClass("previousOverlay nextOverlay");
set.addClass("currentOverlay");
for(i=n; i<config.macros.slideShow.numOverlays; i++){
set = $(".viewer "+"."+cm.text.overlay+(i+1));
set.removeClass("previousOverlay currentOverlay");
set.addClass("nextOverlay");
}
cm.curOverlay = n;
}
config.macros.slideShow.firstSlide = function(){
config.macros.slideShow.showSlide(1);
return false;
}
config.macros.slideShow.lastSlide = function(){
config.macros.slideShow.showSlide(config.macros.slideShow.nSlides);
return false;
}
config.macros.slideShow.next = function(){
var cm = config.macros.slideShow;
if(!cm.overlays || cm.numOverlays == 0 || cm.curOverlay == cm.numOverlays) {
cm.showSlide(cm.curSlide+1);
} else {
cm.showOverlay(cm.curOverlay+1);
}
return false;
}
config.macros.slideShow.previous = function(){
var cm = config.macros.slideShow;
if(!cm.overlays || cm.numOverlays == 0 || cm.curOverlay == 0) {
cm.showSlide(cm.curSlide-1);
cm.showOverlay(cm.numOverlays);
} else {
cm.showOverlay(cm.curOverlay-1);
}
return false;
}
config.macros.slideShow.endSlideShow=function(){
var cm = config.macros.slideShow;
if(cm.autoAdvance) {clearInterval(cm.autoAdvance);}
if(cm.clock) clearInterval(cm.slideClock);
story.closeAllTiddlers();
cm.toggleSlideStyles();
story.displayTiddlers(null,cm.openTiddlers);
$(document).unbind();
document.oncontextmenu = function(){};
$("body").removeClass("slide"+cm.curSlide);
return false;
}
// 'keys' code adapted from S5 which in turn was adapted from MozPoint (http://mozpoint.mozdev.org/)
config.macros.slideShow.keys = function(key) {
var cm = config.macros.slideShow;
switch(key.which) {
case 32: // spacebar
if(cm.auto>0 && cm.blocked==0){
if(cm.autoAdvance){
clearInterval(cm.autoAdvance);
cm.autoAdvance = null;
} else {
cm.autoAdvance=setInterval(cm.next, cm.auto*1000);
}
} else {
if(cm.blocked==0) cm.next();
}
break;
case 34: // page down
if(cm.blocked==0) cm.showSlide(cm.curSlide+1);
break;
case 39: // rightkey
if(cm.blocked==0) cm.next();
break;
case 40: // downkey
if(cm.blocked==0) cm.showOverlay(cm.numOverlays);
break;
case 33: // page up
if(cm.blocked==0) cm.showSlide(cm.curSlide-1);
break;
case 37: // leftkey
if(cm.blocked==0) cm.previous();
break;
case 38: // upkey
if(cm.blocked==0) cm.showOverlay(0);
break;
case 36: // home
if(cm.blocked==0) cm.firstSlide();
break;
case 35: // end
if(cm.blocked==0) cm.lastSlide();
break;
case 27: // escape
cm.endSlideShow();
break;
case 66: // B
$("#slideBlanker").toggle();
cm.blocked = (cm.blocked +1)%2;
break;
}
return false;
}
config.macros.slideShow.clicker = function(e) {
var cm = config.macros.slideShow;
if(cm.blocked==1 || $(e.target).attr('href') || $(e.target).parents().andSelf().hasClass('noClicks')){
return true;
}
if($("#toc").is(':visible')){
cm.toggleTOC();
} else {
if((!e.which && e.button == 1) || e.which == 1) {
cm.next();
}
if((!e.which && e.button == 2) || e.which == 3) {
cm.previous();
}
}
return false;
}
config.macros.slideShow.setClock = function(){
var cm = config.macros.slideShow;
var actualTime = new Date();
var newTime = actualTime.getTime() - cm.clockStartTime;
newTime = cm.clockMultiplier*newTime+cm.clockInterval+cm.clockCorrection;
actualTime.setTime(newTime);
newTime = actualTime.formatString(cm.clockFormat);
$("#slideClock").text(newTime);
return false;
}
config.macros.slideShow.resetClock = function(){
var cm = config.macros.slideShow;
if(cm.clock == 0) return;
var time = new Date(0);
if(cm.clockStartTime>time){
var startTime = new Date();
cm.clockStartTime=startTime.getTime();
}
return false;
}
config.shadowTiddlers.SlideShowStyleSheet="/*{{{*/\n.header, #mainMenu, #sidebar, #backstageButton, #backstageArea, .toolbar, .title, .subtitle, .tagging, .tagged, .tagClear, .comment{\n display:none !important\n}\n\n#slideBlanker{\n position: absolute;\n top: 0;\n left: 0;\n width: 100%;\n height: 100%;\n z-index: 90; \n background-color: #000;\n opacity: 0.9;\n filter: alpha(opacity=90)\n}\n\n.nextOverlay{\n visibility: hidden\n}\n\n.previousOverlay,.currentOverlay{\n visibility: visible\n}\n\n#displayArea{\n font-size: 250%;\n margin: 0 !important;\n padding: 0\n}\n\n#controlBar{\n position: fixed;\n bottom: 2px;\n right: 2px;\n width: 100%;\n text-align: right\n}\n\n#controlBar .button{\n margin: 0 0.25em;\n padding: 0 0.25em\n}\n\n#slideHeader{\n font-size: 200%;\n font-weight: bold\n}\n\n#slideFooter{\n position: fixed;\n bottom: 2px\n}\n\n.slideFooterOff #navigator{\n visibility: hidden\n}\n\n#slideClock{\n margin: 0 5px 0 5px\n}\n\n#slideCounter{\n cursor: pointer;\n color: #aaa\n}\n\n#toc{\n display: none;\n position: absolute;\n font-size: .75em;\n bottom: 2em;\n right: 0;\n background: #fff;\n border: 1px solid #000;\n text-align: left\n}\n\n#jumpItem{\n padding-left:0.25em\n}\n\n#jumpInput{\n margin-left: 0.25em;\n width: 3em\n}\n\n.tocLevel1{\n font-size: .8em\n}\n\n.tocLevel2{\n margin-left: 1em;\n font-size: .75em\n}\n\n.tocLevel3{\n margin-left: 2em;\n font-size: .7em\n}\n\n.tocLevel4{\n margin-left: 3em;\n font-size: .65em\n}\n\n.tocLevel5{\n margin-left: 4em;\n font-size: .6em\n}\n\n.tocLevel6{\n margin-left: 5em;\n font-size: .55em\n}\n/*}}}*/";
config.shadowTiddlers.SlideShowPluginDoc="The documentation is available [[here|http://www.math.ist.utl.pt/~psoares/addons.html#SlideShowPluginDoc]].";
})(jQuery)
}
//}}}
Now we have the definition of the algebraic structure associated to classical field theory, one can ask whether there is a quantization. A definition of quantum field theory on a manifold $ M$ is given in \cite{webbook}. There, an obstruction-theoretic framework is described for analyzing quantizations of a classical field theory.
We will show that any quantum field theory on $ M $ in the sense of \cite{webbook} can be turned into a factorization algebra on $ M $.
''Theorem''
//Let us fix a classical field theory on $M$, with space of fields $\mathcal {E}$ and local action functional $ S $. Let $\Obs^{cl}(\mathcal {E},S)$ be the associated factorization algebra. Thus, $\Obs^{cl}(\mathcal {E},S)$ is a $P_0$ algebra in the category of factorization algebras on $M$.//
//Then, a quantization of the classical field theory described by $ S $ to a quantum field theory, in the sense of \cite{webbook}, yields a quantization of the $ P_0 $ factorization algebra $\Obs^{cl}(\mathcal {E},S)$ into a $ BD $ factorization algebra.//
//Remark//: we do not show that //all// quantizations of the $ P_0 $ factorization algebra $\Obs^{cl}(\mathcal {E},S)$ arise in this way. However, we conjecture that this is the case.
!!! Obstruction theory
It is worth describing the obstruction theory developed in \cite{webbook}. The space $\mathcal{O}_l(\mathcal {E})$ of local action functionals on $\mathcal {E}$ -- that is, of possible Lagrangians on the space of fields $\mathcal {E}$ -- is a cochain complex in a natural way. In the ~Batalin-Vilkovisky picture, the differential is given by bracketing with the classical action $S$.
The zeroth cohomology, $H^0 (\mathcal{O}_l(\mathcal {E}))$, describes equivalence classes of first order deformations of the classical action $S$, satisfying the relevant gauge symmetry conditions. The first cohomology group $H^1 (\mathcal{O}_l(\mathcal {E}))$ describes the obstructions to deforming the classical action in a way preserving the gauge symmetries. The first negative cohomology group $H^{-1} (\mathcal{O}_l(\mathcal {E}))$ describes the infinitesimal automorphisms of the classical field theory described by $(\mathcal {E},S)$. The higher negative cohomology groups correspond to automorphisms of automorphisms, etc.
In \cite{webbook} the following result is proved.
''Theorem.''
//Suppose we have a quantization of the classical field theory defined by $ (\mathcal{E},S) $ to a quantum theory defined modulo $\hbar^{n+1}$. Then the obstruction to lifting this to a quantization defined modulo $\hbar^{n+2}$ is an element of $H^1 (\mathcal{O}_l(\mathcal {E}))$. If this obstruction vanishes, then the space of possible lifts is a torsor for $H^0 (\mathcal{O}_l(\mathcal {E}))$ (more precisely, the simplicial set of possible lifts is a torsor for the simplicial abelian group $H^{\le 0} (\mathcal{O}_l(\mathcal {E}))$).//
This theorem is a mathematical encoding of the renormalization procedure, and is proved using the physicists toolbox of counterterms, Feynman graphs, etc. Together with the previous theorem, this result allows one to produce, starting from a classical Lagrangian, an algebraic object -- a factorization algebra -- which encodes the correlation functions of the quantum field theory. As is usual, there is an ambiguity in going from the classical Lagrangian to the quantum theory; at each order in $\hbar$, we are free to add on an arbitrary term to the classical Lagrangian. However, for //renormalizable// classical Lagrangians, an extra symmetry -- that of rescaling of space-time -- restricts greatly the possible quantizations of a given classical theory (strictly speaking, this symmetry only holds up to terms logarithmic in the scale). This point of view on renormalizability is detailed in \cite{webbook}. In good cases, the space of possible quantizations is parametrized by a finite number of coupling constants.
Pure ~Yang-Mills theory fits into this class of theories. In \cite{webbook}, the deformation-obstruction complex for quantizing pure ~Yang-Mills theory on $\mathbb{R}^4$ was calculated. It was shown that if we restrict attention to quantizations satisfying an additional scaling symmetry, present by virtue of renormalizability of the classical Lagrangian, then the obstruction group vanishes. Further, if we take coefficients in a simple Lie algebra, the deformation group is $1$ dimensionsal. This means that the moduli space of possible quantizations is isomorphic to $\hbar \mathbb{R} [[\hbar]]$; so that the space of theories is parametrized by a single $\hbar$ dependent coupling constant.
Thus, we find
''Corollary.''
//There is a factorization algebra on any $4$-manifold $M$ with a flat metric, which encodes pure ~Yang-Mills theory on $M$.//
/***
|Name|[[StoryViewerPlugin]]|
|Source|http://www.TiddlyTools.com/#StoryViewerPlugin|
|Documentation|http://www.TiddlyTools.com/#StoryViewerPluginInfo|
|Version|1.4.0|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|view a set of tiddlers using a droplist, "first/previous/next/last" links, or timed slideshow|
The {{{<<storyViewer>>}}} macro allows you to quickly ''display //and// navigate between a set of tiddlers'', using a droplist of titles and/or individual "first/previous/next/last" buttons/text links. It also provides a "slideshow" feature that permits you to ''present one tiddler at a time with a countdown timer to automatically advance to the next tiddler'' after a specified number of seconds.
!!!!!Documentation
> see [[StoryViewerPluginInfo]]
!!!!!Revisions
<<<
2011.03.11 1.4.0 added 'sort:fieldname' parameter
2011.01.24 1.3.4 in droplist onchange handler, don't clear slideshow 'started' flag (allows slideshow to continue after manual navigation)
|please see [[StoryViewerPluginInfo]] for additional revision details|
2007.10.23 1.0.0 Initial release, split {{{<<storyViewer>>}}} macro definition from [[StorySaverPlugin]] to allow separate installation of story saving vs. story viewing features.
<<<
!!!!!Code
***/
//{{{
version.extensions.StoryViewerPlugin= {major: 1, minor: 4, revision: 0, date: new Date(2011,3,11)};
config.macros.storyViewer = {
tag: "story",
storynotfoundmsg: "'%0' is an empty/unrecognized story",
firstcmd: "first",
firstbutton: "<<",
firstmsg: "first: '%0'",
nextcmd: "next",
nextbutton: ">",
nextmsg: "next: '%0'",
previouscmd: "previous",
previousbutton: "<",
prevmsg: "previous: '%0'",
lastcmd: "last",
lastbutton: ">>",
lastmsg: "last: '%0'",
refreshmsg: "redisplay '%0'",
refreshmsg: "",
autostart: false,
handler: function(place,macroName,params,wikifier,paramString,tiddler) {
var parsed=paramString.parseParams('anon',null,true,false,false);
var here=story.findContainingTiddler(place);
if (here) var tid=here.getAttribute("tiddler");
var storyname="";
var p=params.shift();
var keywords=['first','previous','here','next','last','list','links','timer','sort'];
if (!p || keywords.indexOf(p.split(':')[0])!=-1) {
// find story from current tiddler name
if (!tid) return; // not in a tiddler... do nothing!
var stories=store.getTaggedTiddlers(this.tag);
if (!stories) return;
for (var s=0; s<stories.length; s++) {
if (!stories[s].linksUpdated) stories[s].changed();
var tids=stories[s].links.slice(0);
if (tids.contains(tid)) { storyname=stories[s].title; break; }
}
if (!storyname.length) return; // current tiddler is not part of a saved story
}
else { storyname=p; p=params.shift(); } // user-specified story name
var sortby=getParam(parsed,'sort','title');
var tids=this.getStory(storyname,sortby); // get tiddler list
var target=null;
switch (p?p.split(':')[0]:'') {
case 'first':
target=tids[0];
break;
case 'previous':
var i=tids.indexOf(tid);
if (i!=-1) var target=tids[Math.max(i-1,0)];
break;
case 'here':
if (tid) target=tid;
break;
case 'next':
var i=tids.indexOf(tid);
if (i!=-1) var target=tids[Math.min(i+1,tids.length-1)];
break;
case 'last':
target=tids[tids.length-1];
break;
case 'links':
this.renderAllLinks(place,storyname);
break;
case 'timer':
var delay=parseInt(getParam(parsed,'timer',15))*1000; // msecs between slides
var autostart=params[0]=='autostart'; if (autostart) params.shift();
var action=params[0]; // null/close/fold
this.renderTimer(place,tids,tid,delay,autostart,action);
break;
case 'list':
default:
var prompt=getParam(parsed,'prompt',storyname+'...');
var nobuttons=params.contains("nobuttons");
var allbuttons=params.contains("allbuttons");
var onlybuttons=params.contains("onlybuttons");
this.renderList(place,tids,tid,storyname,prompt,nobuttons,allbuttons,onlybuttons);
break;
}
var label=getParam(parsed,'label',params[0]||target);
if (target) this.renderLink(place,tid,target,label);
},
getStory: function(storyname,sortby) { // READ TIDDLER LIST
var tids=[];
var fn=store.getMatchingTiddlers||store.getTaggedTiddlers;
var tagged=store.sortTiddlers(fn.apply(store,[storyname]),sortby||'title');
if (tagged.length) // if storyname is a tag, get tagged tiddlers rather than links
for (var t=0; t<tagged.length; t++) tids.push(tagged[t].title);
else {
var t=store.getTiddler(storyname);
if (t && !t.linksUpdated) t.changed();
var tids=t?t.links.slice(0):[];
}
return tids;
},
renderLink: function(place,tid,target,label) {
// override default labelling with specified text (if any)
if (tid==target) { // self-referential links turn into 'refresh links'
var btn=createTiddlyButton(place,null,this.refreshmsg.format([tid]), function() {
var here=story.findContainingTiddler(place).getAttribute("tiddler");
story.refreshTiddler(here,null,true);
});
wikify(label,btn);
}
else // create link
wikify(label,createTiddlyLink(place,target,false));
},
renderAllLinks: function(place,storyname) {
var out="{{floatleft{";
out+="<<storyViewer [["+storyname+"]] first first>> ";
out+="<<storyViewer [["+storyname+"]] previous previous>> ";
out+="}}}";
out+="{{floatright{";
out+=" <<storyViewer [["+storyname+"]] next next>>";
out+=" <<storyViewer [["+storyname+"]] last last>>";
out+="}}}";
out+="{{center{<<storyViewer [["+storyname+"]] here>>}}}";
wikify(out,place);
},
renderList: function(place,tids,tid,storyname,prompt,nobuttons,allbuttons,onlybuttons) {
var h="";
h+='<form style="display:inline">';
if ((!nobuttons||onlybuttons) && allbuttons) {
h+='<input type="button" value="'+this.firstbutton+'" ';
h+=' style="padding:0" title="'+(tids[0]?this.firstmsg.format([tids[0]]):'')+'"';
h+=' onclick="if (this.form.list.length<2) return; ';
h+=' this.form.list.selectedIndex=1; this.form.list.onchange();">';
}
if (!nobuttons||onlybuttons) {
h+='<input type="button" value="'+this.previousbutton+'" style="padding:0 0.3em"';
h+=' onclick="if (this.form.list.length<2) return; ';
h+=' var i=this.form.list.selectedIndex-1; if (i<1) i=1; ';
h+=' this.form.list.selectedIndex=i; this.form.list.onchange();"';
h+=' onmouseover="if (this.form.list.length<2) return; ';
h+=' var i=this.form.list.selectedIndex-1; if (i<1) i=1; ';
h+=' var v=this.form.list.options[i].value; if (!v.length) return; ';
h+=' this.title=config.macros.storyViewer.prevmsg.format([v]);">';
}
h+='<select size="1" name="list"';
if (onlybuttons) h+=' style="display:none;"';
h+=' onchange="if (this.value) story.displayTiddler(this,this.value);">';
h+='<option value="">'+prompt+'</option>';
for (i=0; i<tids.length; i++) {
h+='<option '+
(tids[i]==tid?'selected ':'')+
'value="'+tids[i]+'">\xa0\xa0'+tids[i]+'</option>';
}
h+='</select>';
if (!nobuttons||onlybuttons) {
h+='<input type="button" value="'+this.nextbutton+'" style="padding:0 0.3em"';
h+=' onclick="var i=this.form.list.selectedIndex+1; ';
h+=' if (i>this.form.list.options.length-1) i=this.form.list.options.length-1; ';
h+=' this.form.list.selectedIndex=i; this.form.list.onchange();"';
h+=' onmouseover="var i=this.form.list.selectedIndex+1; ';
h+=' if (i>this.form.list.options.length-1) i=this.form.list.options.length-1; ';
h+=' var v=this.form.list.options[i].value; if (!v.length) return;';
h+=' this.title=config.macros.storyViewer.nextmsg.format([v]);">';
}
if ((!nobuttons||onlybuttons) && allbuttons) {
h+='<input type="button" value="'+this.lastbutton+'" ';
h+=' style="padding:0" title="'+(tids[tids.length-1]?this.lastmsg.format([tids[tids.length-1]]):'')+'"';
h+=' onclick="this.form.list.selectedIndex=this.form.list.options.length-1; this.form.list.onchange();">';
}
h+='</form>';
createTiddlyElement(place,"span").innerHTML=h;
},
renderTimer: function(place,tids,tid,delay,autostart,action) {
var now=new Date().getTime(); // msec
var target=createTiddlyElement(null,'input',now+Math.random()); // unique ID
target.setAttribute('type','button'); target.style.padding='0';
place.appendChild(target);
target.tid =tids[Math.min(tids.indexOf(tid)+1,tids.length-1)]||''; // next tiddler
target.action =action;
target.formatTimer =this.formatTimer;
target.start =this.startTimer;
target.stop =this.stopTimer;
target.onmouseover =this.pauseTimer;
target.onmouseout =this.resumeTimer;
target.tick =this.timerTick;
target.onclick =this.timerClick;
target.next =this.timerNext;
target.start(delay,autostart);
},
formatTimer: function(t) {
return '0:'+String.zeroPad(Math.floor(t/1000),2);
},
startTimer: function(delay,start) {
var co=config.options; // abbrev
start=config.macros.storyViewer.started=start||config.macros.storyViewer.started;
var now=new Date().getTime(); // msec
this.started=start;
this.delay=delay;
this.paused=start?0:delay;
this.stopTime=now+delay; // msec
this.title='CLICK='+(start?'reset':'start')+" slideshow timer... next: '"+this.tid+"'";
this.style.cursor='pointer';
this.value=this.formatTimer(delay);
if (start) {
var code="var e=document.getElementById('"+this.id+"'); if(e)e.tick()";
this.timer=setTimeout(code,500);
}
return false;
},
stopTimer: function() {
this.timer=clearTimeout(this.timer);
this.started=config.macros.storyViewer.started=false;
this.paused=0;
this.title="CLICK=start slideshow timer... next: '"+this.tid+"'";
this.value=this.formatTimer(this.delay);
return false;
},
pauseTimer: function() {
if (!this.started) return;
var now=new Date().getTime(); // msec
this.paused=Math.max(this.stopTime-now,0);
this.stopTime=now+this.paused;
return false;
},
resumeTimer: function() {
if (!this.started || !this.paused) return;
var now=new Date().getTime(); // msec
this.stopTime=now+this.paused;
this.paused=0;
return false;
},
timerTick: function() {
var now=new Date().getTime(); // msec
if (!this.started)
this.stopTime=now+this.delay;
else if (this.paused) {
this.stopTime=now+this.paused;
this.title="[PAUSED] MOUSEOUT=resume, CLICK=reset... next: '"+this.tid+"'";
}
var remaining=this.stopTime-now;
if (remaining>0) {
if (this.started && !this.paused) this.value=this.formatTimer(remaining);
var code="var e=document.getElementById('"+this.id+"'); if(e)e.tick()";
this.timer=setTimeout(code,500);
} else {
this.stop();
this.next();
}
return false;
},
timerClick: function() {
return this.started?this.stop():this.start(this.delay,true);
},
timerNext: function() { // OPEN NEXT TIDDLER
var here=story.findContainingTiddler(this);
config.macros.storyViewer.started=true; // next slide autostarts to continue slideshow
if (this.tid) story.displayTiddler(here,this.tid);
config.macros.storyViewer.started=false;
if (!here) return false;
var t=here.getAttribute('tiddler');
if (this.action=='close') story.closeTiddler(t);
if (this.action=='fold' && config.commands.collapseTiddler) // see CollapseTiddlerPlugin
config.commands.collapseTiddler.handler(null,here,t);
return false;
}
}
//}}}
//{{{
config.paramifiers.story = {
onstart: function(v) {
var t=store.getTiddler(v); if (t) t.changed();
var list=t?t.links:store.getTiddlerText(v,"").parseParams("open",null,false);
story.displayTiddlers(null,list);
}
};
//}}}
In this page we will define what it means to have a factorization algebra endowed with the structure of an algebra over an operad. Since, in this paper, we are principally concerned with factorization algebras taking values in the category of cochain complexes (of nuclear spaces) we will restrict attention to this case in the present page.
Not all operads work for this construction: only operads endowed with an extra structure -- that of a //Hopf operad// can be used.
''Definition''. A Hopf operad is an operad in the category of differential graded cocommutative coalgebras.
Any Hopf operad $P$ is, in particular, a differential graded operad. In addition, the cochain complexes $P(n)$ are endowed with the structure of differential graded commutative coalgebra. The operadic composition maps
$$
\circ_i : P(n) \otimes P(m) \to P(n+m-1)
$$
are maps of coalgebras, as are the maps arising from the symmetric group action on $P(n)$.
If $P$ is a Hopf operad, then the category of dg $P$-algebras becomes a symmetric monoidal category. If $A,B$ are $P$-algebras, the tensor product $A \otimes_{\C} B$ is also a $P$-algebra. The structure map
$$
P_{A \otimes B} : P(n) \otimes (A \otimes B)^{\otimes n} \to A\otimes B
$$
is defined to be the composition
$$
P(n) \otimes (A \otimes B)^{\otimes n} \xto{c(n)} P(n) \otimes P(n) \otimes A^{\otimes n} \otimes B^{\otimes n}
\xto{P_A \otimes P_B} A \otimes B.
$$
In this diagram, $c(n) : P(n) \to P(n)^{\otimes 2}$ is the comultiplication on $c(n)$.
Any operad arising from an operad in topological spaces is a Hopf operad (because topological spaces are automatically cocommutative coalgebras, with comultiplication defined by the diagonal map). For example, the commutative operad $\op{Com}$ is a Hopf operad, with coproduct defined on the generator $\star \in \op{Com}(2)$ by
$$
c ( \star ) = \star \otimes \star.
$$
With the comultiplication defined in this way, the tensor product of commutative algebras is the usual one. If $A$ and $B$ are commutative algebras, the product on $A \otimes B$ is defined by
$$
(a \otimes b) \star (a' \otimes b') = (-1)^{\abs{a'} \abs{b} }(a \star a') \otimes (b \star b').
$$
The Poisson operad is also a Hopf operad, with coproduct defined (on the generators $\star, \{-,-\}$ by
$$
\begin{split}
c( \star) &= \star \otimes \star \\
c ( \{-,-\} ) &= \{-,-\} \otimes \star + \star \otimes \{-,-\}.
\end{split}
$$
If $A$, $B$ are Poisson algebras, then the tensor product $A \otimes B$ is a Poisson algebra with product and bracket defined by
$$
\begin{split}
(a \otimes b) \star (a' \otimes b') &= (-1)^{\abs{a'} \abs{b}} (a \star a' ) \otimes (b \star b') \\
\{a \otimes b, a' \otimes b'\} &= (-1)^{\abs{a'} \abs{b} }\left( \{a,a'\} \otimes (b \star b') + (a \star a') \otimes \{b, b'\} \right) .
\end{split}
$$
!!! Structured factorization algebras
''Definition''. Let $P$ be a differential graded Hopf operad. A prefactorization $P$-algebra is a prefactorization algebra with values in the symmetric monoidal category of $P$-algebras. A factorization $P$-algebra is a prefactorization $P$-algebra, such that the underlying prefactorization algebra with values in cochain complexes is a factorization algebra.
We can unpack this definition as follows. Suppose that $\F$ is a factorization $P$-algebra. Then, $\F$ is a factorization algebra; and, in addition, for all $U \subset M$, $\F(U)$ is a $P$-algebra. The structure maps
$$
\F(U_1) \otimes \cdots \otimes \F(U_n) \to \F(V)
$$
(defined when $U_1,\ldots, U_n$ are disjoint open subsets of $V$) are required to be $P$-algebra maps. It is at this point that we need the fact that $P$ is a Hopf operad; we use this structure to give $\F(U_1) \otimes \cdots \otimes \F(U_n)$ the structure of $P$-algebra.
/***
Inspired by k2
!General
***/
/*{{{*/
body {
background: #EDEDED;
}
#contentWrapper{
background: #fff;
border:1px solid #DDD;
margin: 0 auto;
width: 1000px;
padding:0;
}
/*}}}*/
/***
!Links
***/
/*{{{*/
a,
a.tiddlyLink,
a.button,
a.externalLink,
#sidebarOptions .sliderPanel a{
color: #1D65BC;
text-decoration: none;
background: transparent;
border: 0;
}
#sidebarContents .button {
margin:0em 0.2em;
padding:0.2em 0.3em;
display:block;
color: #1D65BC;
text-decoration: none;
background: transparent;
border: 0;
}
#sidebarTabs .button {
margin:0em 0.2em;
padding:0.2em 0.3em;
display:block;
color: #1D65BC;
text-decoration: none;
background: transparent;
border: 0;
}
a:hover,
a.tiddlyLink:hover,
a.button:hover,
a.externalLink:hover,
#sidebarOptions .sliderPanel a:hover
{
border: 0;
color: #1D65BC;
text-decoration: underline;
background:transparent;
}
.button:active {background:#1d65bc; border:0;}
.viewer .button:active, .viewer .marked, .viewer .highlight {
color: #fff !important;
background: #3371a3;
border: 0;
}
/*}}}*/
/***
!Header
***/
/*{{{*/
.gradient {margin-top:20px; background:#3371A3;}
.titleLine{padding: 80px 40px 32px 40px;}
.titleLine a:hover{color:#fff; border-bottom:1px dotted #eee; text-decoration:none;}
.titleLine a{color:#fff; border-bottom:1px dotted #ccc;}
.siteTitle {
font-size: 2.4em;
font-weight: bold;
color:#fff;
}
.siteSubtitle {
font-size: 1.3em;
display: block;
margin: .3em auto 1em;
color:#fff;
}
/*}}}*/
/***
!TopMenu
***/
/*{{{*/
#topMenu br {display:none; }
#topMenu { background: #3371A3; font-size:1.3em; }
#topMenu { padding:5px 32px; }
#topMenu .button, #topMenu .tiddlyLink {
margin-left:0.1em; margin-right:0.1em;
padding:0.5em;
color:white; font-weight:normal;
}
#topMenu a.button:hover, #topMenu a.tiddlyLink:hover { background:#fff; color:#333; text-decoration:none;}
.searchBar {float:right; font-size:0.9em;}
.searchBar .button {display:block; border:none; color:#ccc;}
.searchBar .button:hover{border:none; color:#eee;}
.searchBar input{
border: 1px inset #1d65bc; background:#dbdee3;
}
.searchBar input:focus {
border: 1px inset #3371a3; background:#fff;
}
/*}}}*/
/***
!Display
***/
/***
!!!Display General
***/
/*{{{*/
#displayArea { margin: 0em 15.7em 0em 1em; }
#displayFooter {
clear: both;
}
#displayFooter {
clear: both;
}
#tiddlerDisplay{padding-top:1em;}
/*}}}*/
/***
!!!Tiddler
***/
/*{{{*/
.tiddler {margin-bottom:1em; padding-bottom:1em;}
.tiddler {padding-left:2em;}
.title {color:#333; font-size:1.8em; border-bottom:1px solid #333; padding-bottom:0.3px;}
.subtitle { font-size:90%; color:#bbb; padding-left:0.25em; margin-top:0.1em; }
.shadow .title {
color: #aaa;
}
h1,h2,h3,h4,h5 { color: #333; background: transparent; padding-bottom:2px; border-bottom: 1px dotted #666; }
* html .viewer pre {
margin-left: 0em;
}
.viewer hr {
border: 0;
border-top: solid 1px #333;
margin: 0 8em;
color: #333;
}
.tiddler .viewer {font-size:1.3em;}
.viewer a.button {color:#000; border:1px solid #1D65BC; font-weight:bold;}
.viewer a.button:hover{color:#fff; background:#3371a3; text-decoration:none;}
.tagClear {clear:none;}
.toolbar .button {color:#bbb; border:none;}
.toolbar .button:hover, .toolbar .highlight, .toolbar .marked, .toolbar a.button:active {background:transparent; color:#111; border:none; text-decoration:underline;}
.tiddler {border-bottom:3px solid #EEF1F3; padding-bottom:2em; padding-top:0em;}
.title {border-bottom:none; margin-right:8em;}
h1,h2,h3,h4,h5 { color: #333; background: transparent; padding-bottom:2px; border-bottom: none; }
.viewer pre, .viewer code {
border: 1px solid #B2B6BE;
background: #EBEEF1;}
.tagging, .tagged {
border: 1px solid #dbdee3;
background-color: #ebeef1;
}
.selected .tagging, .selected .tagged {
background-color: #dbdee3;
border: 1px solid #B2B6BE;
}
.tagging .listTitle, .tagged .listTitle {
color: #bbb;
}
.selected .tagging .listTitle, .selected .tagged .listTitle {
color: #014;
}
.tagging .button:hover, .tagged .button:hover {
border: none; background:transparent; text-decoration:underline; color:#014;
}
.tagged .highlight, .tagged .marked, .tagged a.button:active {text-decoration:underline; background:transparent; color:#014;}
.tagging .button, .tagged .button {
color:#bbb;
}
.selected .tagging .button, .selected .tagged .button {
color:#014;
}
.viewer blockquote {line-height:1.1em; padding-left:0.4em;margin-left:1.0em; border-left: 3px solid [[ColorPalette::TertiaryDark]]; }
.viewer table {
border: 1px solid #3371a3;
}
.viewer th, thead td {
background: #3371a3;
border: 1px solid #3371a3;
color: #fff;
}
.viewer td, .viewer tr {
border: 1px solid #3371a3;
}
/*}}}*/
/***
!!!Editor
***/
/*{{{*/
* html .editor textarea, * html .editor input {
width: 98%;
}
.editor input, .editor textarea {
border: 1px solid #1d65bc; background:#ebeef1;
}
.editor {padding-top:0.3em;}
.editor textarea:focus, .editor input:focus {
border: 1px inset #3371a3; background:#fff;
}
/*}}}*/
/***
!Sidebar
***/
/*{{{*/
#sidebar{
display:inline;
position:relative;
float:right;
margin-bottom:1em;
width: 16em;
}
#sidebar .tabSelected, #sidebar .tabSected:hover {
color: #000;
background: #dbdee3;
border-top: solid 1px #B2B6BE;
border-left: solid 1px #B2B6BE;
border-right: solid 1px #B2B6BE;
border-bottom:solid 1px #dbdee3 !important;
padding-bottom:1px;
text-decoration:none;
}
#sidebarOptions, #sidebarTabs, #sidebarContents {border-left: 1px solid #B2B6BE;}
#sidebarTabs {border-bottom: 1px solid #B2B6BE;}
#sidebar .tabUnselected, #sidebar .tabUnselected:hover {
color: #F0F3F5;
background: #B2B6BE ;
border: solid 1px #B2B6BE ;
padding-bottom:1px;
}
#sidebarTabs .tabContents {border:none; background:#DBDEE3; }
#sidebarTabs .tabContents {border-top:1px solid #B2B6BE;}
#sidebarTabs .tabContents .tabContents {border-left:1px solid #b2b6be;}
#sidebarOptions .sliderPanel {
background: #EBEEF1; border:none;
}
#sidebarOptions input {
border: 1px solid #1d65bc;
}
#sidebarOptions input:hover, #sidebarOptions input:active, #sidebarOptions input:focus {
border: 1px inset #3371a3;
}
#sidebar {background: #EBEEF1 ; right:0;}
#sidebar .button:active, #sidebar .marked, #sidebar .highlight {color:#014; background:transparent;text-decoration:none}
/*}}}*/
/***
!!Popups
***/
/*{{{*/
.popup {
background: #3371a3;
border: 1px solid #333;
}
.popup hr {
color: #333;
background: #333;
border-bottom: 1px;
}
.popup li.disabled {
color: #333;
}
.popup li a, .popup li a:visited {
color: #eee;
border: none;
}
.popup li a:hover {
background: #3371a3;
color: #fff;
border: none;
text-decoration:underline;
}
/*}}}*/
/***
!!Message Area
***/
/*{{{*/
#messageArea {
border: 2px dashed #3371a3;
background: #dbdee3;
color: #fff;
font-size:90%;
}
#messageArea .button {
color: #1d65bc;
background: #ebeef1;
text-decoration:none;
font-weight:bold;
border:none;
}
#messageArea a.button {color:#1d65bc;}
#messageArea .button:hover {text-decoration:underline;}
/*}}}*/
/***
!!Tabs
***/
/*{{{*/
.viewer .tabSelected, .viewer .tabSelected:hover{
color: #014;
background: #e0e0e0;
border-left: 1px solid #B2B6BE;
border-top: 1px solid #B2B6BE;
border-right: 1px solid #B2B6BE;
}
.viewer .tabUnselected, .viewer .tabUnselected:hover {
color: #fff;
background: #B2B6BE;
}
. viewer .tabContents {
color: #ffffff;
background: #ffffff;
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In this page we will give the definition of an action of an $L_\infty$ algebra on a classical field theory. We will start by saying what it means for an $L_\infty$ algebra to act on an elliptic moduli problem.
Recall that in homotopy theory, to give an action of a group $G$ on an object is the same as to give a family of objects over the classifying space $BG$. There is a similar picture in homotopical algebra: to given an action of an $L_\infty$ algebra $\g$ on some object is the same as to give a family of such objects over $C^\ast(\g)$. We will take this as our definition of action of an $L_\infty$ algebra $\g$ on an elliptic $L_\infty$ algebra. Thus, we need to define what it means to have a family of elliptic $L_\infty$ algebras over some differential graded base ring.
Suppose that $R$ is a differential graded algebra. Let $R^\sharp$ refer to $R$ without the differential.
''Definition.''
//An $R$-family of elliptic $L_\infty$ algebras on $X$ consists of graded bundle $L$ of $R^\sharp$-modules on $X$, whose sheaf of sections will be denoted $\L$; together with an $R^\sharp$-linear differential operator//
$$
\d : \L \to \L
$$
//which makes $\L$ into a sheaf of dg $R$-modules; and, collection of $R$-linear polydifferential operators//
$$
l_n : \L^{\otimes n} \to \L
$$
//making $\L$ into a sheaf of $L_\infty$ algebras on $X$ over $R$. //
//Remark:// Note that in this definition, $R$ can be a nuclear Frechet dg algebra. In that case, the tensor products should be the completed projective tensor product.
''Definition.''
//If $\g$ is an $L_\infty$ algebra, and $\mscr{L}$ is an elliptic $L_\infty$ algebra on a space $X$, a $\g$-action on $\mscr{L}$ is a family of elliptic moduli problems $\mscr{L}^{\g}$ on $X$, over the base ring $C^\ast(\g)$, which specialize to $\mscr{L}$ modulo the maximal ideal $C^{>0}(\g)$ of $C^\ast(\g)$. //
//Remark:// The ~Chevalley-Eilenberg cochain complex $C^\ast(\g)$ is the completed pro-nilpotent dg algebra, which is an inverse limit
$$
C^\ast(\g) = \liminv C^\ast(\g) / I^n
$$
where $I$ is the maximal ideal $C^{>0}(\g)$.
Let $R$ be a differential graded algebra, and let $\mscr{L}$ be an $R$-family of elliptic $L_\infty$ algebras. Recall that this means that we have a graded bundle $L$ of $R^\sharp$-modules on $X$, whose sheaf $\mscr{L}$ of sections is equipped with a differential making it into a sheaf of dg $R$-modules, and with an $R$-linear $L_\infty$ structure. We will let
$$
L^! = L^\vee \otimes \op{Dens}_X
$$
where $L^\vee$ is the $R^\sharp$-linear dual of $L$. We will let $\mscr{L}^!$ denote the sheaf of sections of $L^!$. This has a natural structure of sheaf of dg modules over $R$, with an $L_\infty$ action of $\mscr{L}$.
''Definition.''
//An invariant pairing of degree $k$ on an $R$-family of elliptic $L_\infty$ algebras $\mscr{L}$ is an $R$-linear isomorphism //
$$
\mscr{L} \iso \mscr{L}^! [k]
$$
//of sheaves of $\mscr{L}$-modules, which is symmetric as before.//
''Definition.''
//Let $\g$ be an $L_\infty$ algebra, and let $\mscr{L}$ be a classical field theory on a space $X$. Thus $\mscr{L}$ is an elliptic $L_\infty$ algebra on $X$ with an invariant pairing $\mscr{L} \iso \mscr{L}^![-3]$ of degree $-3$. Then a $\g$-action on $\mscr{L}$ is a family of elliptic moduli problems $\mscr{L}^{\g}$ on $X$, flat over the base ring $C^\ast(\g)$, equipped with an invariant pairing of degree $-3$, which specializes to $\mscr{L}$ modulo the maximal ideal $C^{>0}(\g)$ of $C^\ast(\g)$. //
If $\mscr{L}$ is an elliptic $L_\infty$ algebra on $X$ with an action of $\g$, then the cotangent field theory $T^\ast[-1] \mscr{L}$ also has a natural action of $\g$, compatible with the invariant pairing.
!!! Symmetries and local functionals
Let $L$ be a classical field theory, so that $L$ is a local $L_\infty$ algebra on $M$ equipped with a non-degenerate symmetric isomorphism $L \iso L^![-3]$. [[Recall |definition of classical field theory]] that the $L_\infty$ structure on $L$ can be described by a local functional $S \in \Ool(B \L)$, satisfying the classical master equation $\{S,S\} = 0$.
We will use this presentation to show that the dg Lie algebra controlling symmetries and deformations of the classical field theory $L$ is $\Ool(B \L)[-1]$, with the bracket $\{-,-\}$ and differential $\{S,-\}$.
More precisely, we will verify the following.
''Proposition''
//Let $\g$ be an $L_\infty$ algebra. Then every $\g$ action on $L$ (in the sense described above) arises from an $L_\infty$ map $\g \to \Ool(B\L)[-1]$. This $L_\infty$ map is unique up to homotopy. //
''Proof.''
An $L_\infty$ map $\g \to \Ool(B \L)[-1]$ is the same as an element
$$\alpha \in C^\ast(\g) \otimes \Ool(B \L)[-1]$$
which
# Satisfies the ~Maurer-Cartan equation.
# Vanishes modulo the maximal ideal $C^{>0} (\g)$ of $C^\ast(\g)$.
The dg Lie algebra structure on $C^\ast(\g) \otimes \Ool(B \L)[-1]$ arises from the dg commutative algebra structure on $C^\ast(\g)$ and the dg Lie algebra structure on $\Ool(B \L)[-1]$.
The ~Maurer-Cartan equation is, of course,
$$
\d_\g \alpha + \{S,\alpha\} + \tfrac{1}{2}{\alpha,\alpha} = 0
$$
where $\d_\g$ refers to the differential on $C^\ast_{red}(\g)$.
Given such an $\alpha$, it is immediate that $\alpha + S$ gives $C^\ast(\g) \otimes L$ the structure of a family of classical field theories over $C^\ast(\g)$, as desired.
Conversely, suppose that we have a family of classical field theories over $C^\ast(\g)$, whose space of fields is the sections of some bundle $L'$ of $C^\ast(\g)$-modules $L'$ on $M$. Let us suppose that this family reduces modulo the maximal ideal $C^{>0}(\g)$ of $C^\ast(\g)$ to the given classical field theory, with underlying vector bundle $L$ on $M$ and action functional $S \in \Ool(B \L)$. Then, it is easy to verify that we can choose an isomorphism of graded $C^\sharp(\g)$-modules $L' \iso C^\sharp(\g) \otimes L$. The structure of family of classical field theories on sections of $L'$ is encoded in some $\phi \in C^\ast(\g) \otimes \Ool(B \L)[-1]$. Modulo the maximal ideal of $C^\ast(\g)$, $\phi$ must reduce to the given action functional $S$, so that $\phi = S + \alpha$ where $\alpha$ is as above.
The final thing we need to verify is the statement that the $L_\infty$ map $\g \to \Ool(B \L)[-1]$ corresponding to a given family of classical field theories over $C^\ast(\g)$ is unique up to homotopy. The $L_\infty$ map is associated to the family of classical field theories together with the isomorphism of graded $C^\sharp(\g)$-modules
$$\Gamma : L' \iso C^\sharp(\g) \otimes L$$
(of course, this isomorphism must be the given one modulo the maximal ideal of $C^\ast(\g)$). The space of such isomorphisms is contractible. Thus, it suffices to verify that if we have a family of such isomorphisms, parametrized by the $n$-simplex, we get a family of $L_\infty$ maps over the base ring $\Omega^\ast(\bigtriangleup^n)$. If
$$
\til{\Gamma} : L' \otimes \cinfty(\bigtriangleup^n) \iso C^\sharp(\g) \otimes L \otimes \cinfty(\bigtriangleup^n)
$$
is such a family, it extends by linearity to an isomorphism of graded $C^\sharp(\g) \otimes \Omega^\sharp(\bigtriangleup^n)$-modules
$$
L' \otimes \Omega^\sharp(\bigtriangleup^n) \iso C^\sharp(\g) \otimes L \otimes \Omega^\sharp(\bigtriangleup^n).
$$
The left hand side is equipped with the structure of a family of classical field theories over $C^\ast(\g) \otimes \Omega^\ast(\bigtriangleup^n)$ (this family is constant in the $\bigtriangleup^n$-directions). As above, this structure can be interpreted as giving a ~Maurer-Cartan element $\alpha$ of $C^\ast(\g) \otimes \Omega^\ast(\bigtriangleup^n) \otimes \Ool(B \L)[-1]$, which vanishes modulo the maximal ideal of $C^\ast(\g)$. Such a ~Maurer-Cartan element is precisely the same as a family of $L_\infty$ maps $\g \to \Ool(B \L)[-1]$ over the base ring $\Omega^\ast(\bigtriangleup^n)$.
$\square$
In this page, as a prelude to giving a formal definition of what we mean by a classical field theory, we will discuss odd symplectic structures on spaces of fields. Such symplectic structures play a key role in the ~Batalin-Vilkovisky formalism.
Let $E$ be a graded vector bundle on $M$.
''Definition''
A degree $-1$ symplectic pairing on $E$ is a vector bundle map
$$
E \otimes E \to \op{Dens}(M)
$$
of cohomological degree $-1$ and antisymmetric, such that the adjoint map
$$
E \to E \otimes \op{Dens}(M)
$$
is an isomorphism of vector bundles on $M$.
If $E$ is a vector bundle on $M$ with a symplectic pairing of degree $-1$, then there is a corresponding pairing
$$
\ip{-,-} : \E_c(U) \otimes \E_c(U) \to \R
$$
on the space $\E_c(U)$ of compactly supported sections of $E$. This pairing is symplectic of cohomological degree $-1$. "Symplectic" means that the pairing is antisymmetric and has no kernel. The pairing does not have an inverse in the space $\E_c(U)$, however.
We can view this symplectic pairing as providing an isomorphism
$$
\E(U) \iso \E^!(U) [1]
$$
where $\E^!(U)$ is, as explained [[here | classes of sections of a vector bundle]] the space of sections of the bundle $E^! = E^\vee \otimes \op{Dens}(M)$ on $U$.
!!! Poisson brackets
We would like to consider the corresponding Poisson bracket on functions on $\E(U)$. This Poisson bracket will not be defined on all functions.
The symplectic pairing on $E$ gives an isomorphism $\E^!(U) [1] \iso \E(U)$, and thus an isomorphism
$$
\Oo(\E_c(U)) \otimes \E_c^!(U) \iso \op{Der}(\Oo(\E_c(U) ) [-1],
$$
where $\op{Der}(\Oo(\E_c(U))$ denotes the [[derivations]] of $\Oo(\E_c(U))$.
If $\Phi \in \Oo^{P, sm}(\E_c(U))$, then, by assumption
$$
\d \Phi \in \Oo(\E_c(U)) \otimes \E^!(U) \iso \Oo(\E_c(U)) \otimes \E(U)[-1] \iso \op{Der}(\Oo(\E(U)) [-1].
$$
We will let $X_\Phi$ be the derivation corresponding to $\d \Phi$. Note that it has cohomological degree $1$.
Now, if $\Phi \in \Oo^{P, sm}(\E_c(U))$ and $\Psi \in \Oo(\E(U))$ is an arbitrary function, we can define
$$
\{\Phi,\Psi\} = X_\Phi \Psi \in \Oo(\E(U)).
$$
Note that
$$
\op{deg} \{\Phi,\Psi\} = \op{deg} \Phi + \op{deg} \Psi + 1.
$$
Further, this Poisson bracket does not define a continuous bilinear map
$$
\Oo^{P, sm}(\E_c(U)) \times \Oo(\E(U)) \to \Oo(\E(U)).
$$
Thus, it does not extend to the completed projective tensor product.
But, as with the Lie bracket on [[derivations]], this Poisson bracket is a separately continuous bilinear map, and thus defines a map from the [[completed inductive tensor product]]
$$
\Oo^{P, sm}(\E_c(U)) \br{\otimes} \Oo(\E(U)) \to \Oo(\E(U)).
$$
''Lemma''
For $\Phi \in \Oo^{sm,P}(\E_c(U))$, the derivation
$$X_\Phi : \Oo(\E(U)) \to \Oo(\E(U))$$
extends (uniquely) to a linear map
$$\Oo^{sm,P}(\E_c(U)) \to \Oo^{sm,P}(\E_c(U)).$$
Further, the Poisson bracket $\{-,-\}$ on $\Oo^{sm,P}(\E_c(U))$ satisfies the Jacobi identity, making $\Oo^{sm,P}(\E_c(U))$ into a Lie algebra with a Lie bracket of cohomological degree $1$.
Of course, $\Oo^{sm,P}(\E_c(U))$ is not a Lie algebra in the category of nuclear spaces with the projective tensor product, but with the completed inductive tensor product.
''Lemma''
* If
$$\Phi,\Psi \in \Oo^{sm}(\E(U)) \subset \Oo^{P,sm}(\E_c(U))$$
then so is $\{\Phi,\Psi\}$.
* The space of local action functionals $\Oo_l(\E_c(U))$ is a subspace of $\Oo^{P,sm}(\E_c(U))$, closed under the Poisson bracket.
* Further, a functional $\Phi \in \Oo^{sm}(\E_c(U))$ is a local action functional if and only if the Taylor components of $\d \Phi$, which are maps
$$
\E_c(U)^{\otimes n} \to \E^!_c(U),
$$
are [[polydifferential operators]].
!!! Derived Euler-Lagrange equations and symplectic structures
We can rewrite the [[derived Euler-Lagrange equations]] using the language introduced in this page. Suppose, as [[before|Euler-Lagrange]], that we are interested in a field theory whose space $\Fields(U)$ (on an open subset $U \subset M$) is the space of sections of a vector bundle $V$ on $U$. Let $S \in \Ool(\Fields_c(M))$ be a local action functional.
Then, the [[derived Euler-Lagrange equations]] for $S$ on $U$ are described by the formal dg manifold constructed from $\Fields(U) \oplus \Fields^!(U)[-1]$. If we let
$$
E = V \oplus V^![-1]
$$
then
$$
\E(U) = \Fields(U) \oplus \Fields^!(U)[-1].
$$
Further, the local action functional $S \in \Ool(\Fields_c(M))$ can be extended to a local action functional (still called $S$) in $\Oo_l(\E_c(U))$.
Further, the Poisson bracket of $S$ with itself is zero. Thus, we can define a differential
$$
\begin{split}
\Oo(\E(U)) & \to \Oo(\E(U)) \\
\Phi \to \{S,\Phi\}.
\end{split}
$$
''Lemma''
The differential graded algebra $\left( \Oo(\E(U)), \{S,-\} \right)$ describes the derived space of solutions to the Euler-Lagrange equations for the local action functional $S$ on $\Fields(U)$. That is, the isomorphism
$$
\Oo(\E(U)) \iso \Oo(\Fields(U) \oplus \Fields^!(U)[-1] )
$$
takes $\{S,-\}$ to the differential described [[earlier|derived Euler-Lagrange equations]].
<<tiddler [[Table of contents]]>>
# [[ Introduction, overview, and physical motivation|Introduction]]
## [[ Factorization algebras in quantum mechanics|The motivating example of quantum mechanics]]
## [[A preliminary definition of prefactorization algebras]]
## [[Prefactorization algebras in quantum field theory]]
# The Main Theorems and a Guide to the Paper
## [[Classical field theory and factorization algebras]]
## [[Quantum field theory and factorization algebras]]
## [[The weak quantization theorem]]
## [[The strong quantization theorem]]
# Factorization algebras and basic examples
## Definition of a [[prefactorization algebra |Prefactorization algebras]]
## [[Concrete examples of prefactorization algebras]]
## Definition of a [[factorization algebra |Factorization algebra]]
## [[Factorization algebras from cosheaves]]
## [[Locally constant factorization algebras]] and $E_n$ algebras
## [[Ordinary quantum mechanics as a factorization algebra]]
# Properties of the category of factorization algebras
## [[The category of factorization algebras]]
## [[Pushforward]]
## [[Extension from a basis]]
## [[Pullback]]
## [[Descent]]
# Operads and factorization algebras
## [[Structured factorization algebras]]
## The [[P_0 operad]]
## The [[Beilinson-Drinfeld operad]]
## [[Lax algebras over an operad]]
# Classical field theory
## [[Motivational overview|Introduction to classical field theory]]
## [[Elliptic moduli problems and local Lie algebras]]
### [[Formal moduli problems and Lie algebras]]
### [[Elliptic moduli problems]]
### [[Examples of elliptic moduli problems related to scalar field theories]]
### [[Examples of elliptic moduli problems related to gauge theories]]
### [[Cochains of a local Lie algebra]]
### [[D-modules and local Lie algebras]]
## Definition of a classical field theory
### [[The classical BV formalism in finite dimensions]]
### [[The classical BV formalism in infinite dimensions]]
### [[The exterior derivative of a local action functional]]
### [[Field theories from action functionals]]
### [[A succinct definition of a classical field theory|Definition of classical field theory]]
## Examples of field theories
### [[Examples of field theories from action functionals]]
### [[Cotangent field theories]]
## [[The graded Poisson structure on classical observables]]
### [[The Poisson structure for free field theories]]
### [[The Poisson structure for cotangent field theories]]
### [[The Poisson structure for a general classical field theory]]
## Symmetries and conserved quantities
### [[Symmetries and deformations of a classical field theory]]
### [[Symmetries and local functionals]]
###[[Conserved quantities and Noether's theorem]]
# Quantum field theory
## [[Outline of the deformation quantization picture]]
## [[Global observables]]
## [[Informal description of the local observables]]
## [[Parametrices]]
## [[Effective interactions and the quantum master equation using parametrices]]
## [[Global observables using parametrices]]
## [[The factorization algebra of observables of a quantum field theory |Quantum observables]]
### [[Local quantum observables]]
### [[The prefactorization algebra of observables]]
### [[The factorization algebra of observables]]
## [[Correlation functions]]
## [[The sheaf of theories|sheaf-of-theories]]
# Examples of ~QFTs
## [[The Weyl algebra and the free field on the real line]]
## [[Free fields]]
### [[The Heisenberg algebra construction]]
### Free fields and [[determinants of complexes|Determinants of complexes]]
## [[The Weyl algebra redux]]
## [[The free holomorphic boson]]
# [[Overview of perturbative quantum field theory]]
## [[Local action functional]]
## [[Quantum field theory]]
## [[BV theory]]
## [[Local-effective correspondence]]
# [[Appendix on homological algebra and functional analysis]]
## [[Classes of sections of a vector bundle]]
## [[Classes of functions on the space of sections of a vector bundle]]
## [[Derivations]]
## [[Nuclear spaces|Overview of nuclear spaces]]
## The [[Completed inductive tensor product]]
## [[Topological cochain complexes]]
## An [[Atiyah-Bott lemma]] for elliptic complexes on open sets
# [[Appendix on dg manifolds and the BV formalism]]
## [[Signs and other conventions]]
## [[Kernels in the BV formalism]]
## [[Differential graded manifolds]]
## [[Derived critical locus]]
/%##[[The Euler-Lagrange equations|Euler-Lagrange]]
## [[Derived Euler-Lagrange equations]]
## [[Gauge theories and nonlinear sigma models in classical field theory]]
## [[Symplectic structures on field spaces]]
## [[ Formal definition|Definition of classical field theory]]
## [[Observables of a classical field theory]]%/
In studying ordinary quantum mechanics for a quadratic action, the Heisenberg algebra plays a useful role. As we will show below, there is a natural appearance of cosheaves of Heisenberg algebras in constructing the factorization algebra of observables.
Recall the usual construction of a Heisenberg algebra. If $V$ is a symplectic vector space, then the Heisenberg algebra $\op{Heis}(V)$ is the central extension
$$
0 \to \C \cdot \hbar \to \op{Heis}(V) \to V
$$
where the bracket is $[x,y] = \hbar \langle x, y \rangle$. Here $V$ consists of the linear observables.
Our construction will also appear through the //linear// observables on a free field. On the field side, we have a vector space with a degree $-1$ symplectic pairing; on the observable side, we have a degree 1 Poisson bracket, which induces a degree 1 pairing on the linear observables. This Poisson bracket leads to a Heisenberg algebra, just as in the usual construction.
!! The basic homological construction
Let $U$ be a graded vector space. Then the sum $V = U \oplus U^\vee[-1]$ has a natural symplectic pairing $\langle -, - \rangle$ of degree -1. Equip $V$ with a differential $d$ that is skew self adjoint for the pairing. This complex $V$ denotes our space of fields. The algebra of functions on $V$ is $\Oo(V) = \what{\Sym}^\ast V^\ast$. The observables are $\Oo(V)[[\hbar]]$ with differential $d + \hbar \Delta$, where $\Delta$ is the BV Laplacian arising from the symplectic pairing.
Consider the linear observables $V^\vee$. The Poisson bracket $\{-,-\}$ on $\Oo(V)$ induces a symplectic pairing of degree 1 on $V^\vee$. The lemma below will justify our interest in working with $W := V^\vee[-1]$, which inherits a symplectic pairing $\{-,-\}$ of degree -1.
''Definition''
The Heisenberg algebra $\op{Heis}(V)$ is the Lie algebra central extension
$$
0 \to \C \cdot \hbar [-1] \to \op{Heis}(V) \to W \to 0
$$
whose bracket is
$$
[ v + \hbar a, w + \hbar b] = \hbar \{ v, w \}.
$$
The element $\hbar$ labels the basis element of the center $\C [-1]$.
Putting the center in degree 1 may look strange, but it is justified for two reasons. First, it makes $\op{Heis}(V)$ into a dg Lie algebra, since the bracket becomes degree 0 through this shift. Second, the following lemma relates this choice to the observables $\Oo(V)$. We will put this relationship to good use in computing factorization homology for the observables of a free field.
Let $\what{C}_\ast(\op{Heis}(V))$ denote the completion of the Lie algebra chain complex of $\op{Heis}(V)$, defined by the product of the spaces $\Sym^n \op{Heis}(V)$, instead of their sum.
''Lemma''
The completed Lie algebra homology complex $\what{C}_\ast(\op{Heis}(V))$ is isomorphic to $\Oo(V)$ with differential $\d + \hbar \Delta$.
//Proof://
The Chevalley-Eilenberg complex for the homology of $\op{Heis}(V)$ has the completed symmetric algebra $\what{\Sym}(\op{Heis}(V)[1])$ as its underlying graded vector space. Note that
$$
\Sym(\op{Heis}(V)[1]) = \Sym(V^\vee \oplus \C \cdot \hbar) = \what{Sym}(V^\vee)[[\hbar]],
$$
so we have the isomorphism as graded vector spaces.
The differential is straightforward as well. On $C_\ast(\op{Heis}(V))$, the differential is a sum of two components: we extend the differential on $\op{Heis}(V)$ by the Leibniz rule and we also acquire a term from the bracket. This second component is exactly the BV Laplacian.$\square$
!! Cosheaves of Heisenberg algebras
For any free theory $\E$, the linear observables $\E^\vee$ of compactly supported distributions forms a cosheaf. The compactly supported smooth sections inside $\E^\vee$ then form a cosheaf. We can apply the construction from above to obtain a cosheaf of Heisenberg algebras, which we denote $\op{Heis}(\E)$, and $\what{C}_\ast(\op{Heis}(\E))$ is precisely the ``smeared" quantum observables $\Obs(\E)$.
This construction is useful because it gives us another way to compute the pushforward of $\Obs(\E)$. Given a smooth map $p: X \to Y$, we can either pushforward $\Obs(\E)$ or we can pushforward the cosheaf $\op{Heis}(\E)$ and then take its Chevalley-Eilenberg homology complex. In other words, pushforward commutes with taking the Lie algebra homology.
In this page we will prove the following.
''Theorem''. //For any [[classical field theory | definition of classical field theory]] on $M$, there is a lax $P_0$ factorization algebra $\til{\Obs}^{cl}$, together with a quasi-isomorphism of factorization algebras//
$$
\til{\Obs}^{cl}(1) \iso \Obs^{cl}.
$$
!!! Functionals with smooth first derivative
As in our discussion of the $P_0$ algebra for [[free field theories|The Poisson structure for free field theories]] and [[cotangent theories|The Poisson structure for cotangent field theories]], we will define a subalgebra $\til{\Obs}^{cl}(U) \subset \Obs^{cl}(U)$ consisting of functionals which have some additional smoothness properties.
Let $L$ be an elliptic $L_\infty$ algebra on $M$, which defines a classical field theory. Recall that the cochain complex of observables is
$$
\Obs^{cl}(U) = C^\ast (\L(U)),
$$
where $\L(U)$ is the $L_\infty$ algebra of sections of $L$ on $U$.
Recall that, as a graded vector space, $C^\ast ( \L(U))$ is the algebra of functionals $\Oo(\L(U)[1])$ on the graded vector space $\L(U)[1]$. In the [[appendix|Classes of functions on the space of sections of a vector bundle]], given any graded vector bundle $E$ on $M$, we define a subspace
$$
\Oo^{sm}(\E(U)) \subset \Oo(\E(U))
$$
of functionals which have "smooth first derivative". Thus, a function $\Phi \in \Oo(\E(U))$ is in $\Oo^{sm}(\E(U))$ if
$$
\d \Phi \in \Oo(\E(U)) \otimes \E_c^!(U).
$$
(The exterior derivative of a general function in $\Oo(\E(U))$ will lie in the larger space $\Oo(\E(U)) \otimes \br{\E}^!_c(U)$).
Recall that, if $\g$ is an $L_\infty$ algebra, the exterior derivative maps $C^\ast(\g)$ to $C^\ast(\g, \g^\vee[-1] )$. The complex $C^\ast_{sm}(\L(U))$ of cochains with smooth first derivative is thus defined to be the subcomplex of $C^\ast(\L(U))$ consisting of this cochains whose first derivative lies in $C^\ast(\L(U) , \L^!_c(U)[-1])$, which is a subcomplex of $C^\ast(\L(U),\otimes \L(U)^\vee[-1])$.
In other words, $C^\ast_{sm}(\L(U))$ is defined by the fibre diagram
$$
\begin{array} { c c c }
C^\ast_{sm}(\L(U)) & \xto{\d} & C^\ast(\L(U), \L_c^!(U)[-1] ) \\
\downarrow & & \downarrow \\
C^\ast(\L(U)) & \xto{\d} & C^\ast(\L(U) , \br{\L_c}^!(U)[-1]).
\end{array}
$$
Note that
$$
C^\ast_{sm}(\L(U)) \subset C^\ast(\L(U))
$$
is a sub-commutative dga, and further, as $U$ varies, $C^\ast_{sm}(\L(U))$ defines a sub-commutative prefactorization algebra of that defined by $C^\ast(\L(U))$.
We will let
$$
\til{\Obs}^{cl}(U) = C^\ast_{sm}(\L(U)) \subset C^\ast(\L(U)) = \Obs^{cl}(U).
$$
!!! The Poisson bracket
Because the elliptic $L_\infty$ algebra $L$ defines a classical field theory, it is equipped with an isomorphism $L \iso L^![-3]$. Thus, we have an isomorphism
$$
\Phi : C^\ast(\L(U), \L_c^!(U)[-1] ) \iso C^\ast(\L(U), \L_c(U)[2] ) .
$$
[[Recall|derivations]] that we can identify $C^\ast(\L(U), \L_c(U)[1])$ with the complex of derivations of $C^\ast(\L(U))$.
Thus, composing the map $\Phi$ with the exterior derivative $\d$, we find a cochain map
$$
C^\ast_{sm}(\L(U)) \to \op{Der} ( C^\ast(\L(U) ) [1].
$$
If $f \in C^\ast_{sm}(\L(U))$ we will let $X_f \in \op{Der} ( C^\ast(\L(U) )$ be the corresponding derivation. If $f$ has cohomological degree $k$, then $X_f$ has cohomological degree $k+1$.
If $f,g \in C^\ast_{sm}(\L(U)) = \til{\Obs}^{cl}(U)$, we define
$$
\{f,g\} = X_f g \in \til{\Obs}^{cl}(U).
$$
This bracket defines a separately continuous bilinear map
$$
\til{\Obs}^{cl}(U) \times \til{\Obs}^{cl}(U) \to \til{\Obs}^{cl}(U).
$$
''Lemma.''
//This bracket satisfies the Jacobi rule and the Leibniz rule. Further, if $U,V$ are disjoint subsets of $M$, both contained in $W$, and if $f \in \til{\Obs}^{cl}(U)$, $g \in \til{\Obs}^{cl}(V)$, then//
$$
\{f,g\} = 0 \in \til{\Obs}^{cl}(W).
$$
''Proof.''
The proof is straightforward.
$\square$
!!! The lax $P_0$ structure
Now, as is explained in the [[appendix|completed inductive tensor product]], a separately continuous bilinear map $E \times F \to G$ between nuclear spaces extends uniquely to a map
$$
E \br{\otimes} F \to G
$$
where $\br{\otimes}$ denotes the [[completed inductive tensor product]].
It follows that $\til{\Obs}^{cl}(U)$ is a $P_0$ algebra in the symmetric monoidal category of topological vector spaces equipped with this inductive tensor product. We will leverage this structure to construct a lax $P_0$ algebra in the symmetric monoidal category of nuclear spaces with the projective tensor product.
Let us write
$$
\Obs^{cl}(U) = \prod_{i \ge 0} \Obs^{cl}_i(U)
$$
where
$$
\Obs^{cl}_i(U) = \Sym^i ( \br{\L}^!_c(U)[-1])
$$
is the space of compactly supported symmetric distributional sections of the bundle $(L^!)^{\boxtimes i}[-i]$ on $U^i$.
Note that the differential maps $\Obs^{cl}_i(U)$ to $\prod_{j \ge i} \Obs^{cl}_j(U)$; and the product maps
$$\Obs^{cl}_i (U) \otimes \Obs^{cl}_j(U) \to \Obs^{cl}_{i+j}(U).$$
We can similarly decompose
$$
\til{\Obs}^{cl}(U) = \prod_i \til{\Obs}_i^{cl}(U) .
$$
Let us define
$$
\til{\Obs}^{cl}(n)(U) = \prod_{i_1,\ldots,i_n} \left(\til{\Obs}_{i_1}^{cl}(U) \bar{\otimes} \cdots \bar{\otimes} \til{\Obs}_{i_n}^{cl}(U) \right).
$$
In particular, $\til{\Obs}^{cl}(1)(U) = \til{\Obs}^{cl}(U)$.
If $V,W$ are any topological vector spaces, there is a natural map $V \br{\otimes} W \to V \otimes W$. Further, since the projective tensor product $\otimes$ commutes with limits, there is a natural map
$$
\til{\Obs}^{cl}(n) (U) \to \Obs^{cl}(U)^{\otimes n}.
$$
The product and Poisson bracket are separately continuous maps
$$
\begin{split}
- \ast- : \til{\Obs}_i^{cl}(U) \times \til{\Obs}_j^{cl}(U) &\to \til{\Obs}_{i+j}^{cl}(U) \\
\{-,-\} : \til{\Obs}_i^{cl}(U) \times \til{\Obs}_j^{cl}(U) &\to \til{\Obs}_{i+j-2}^{cl}(U).
\end{split}
$$
This implies that taking the Poisson bracket or the product on two of the factors gives continuous linear maps
$$
\begin{split}
-\ast-_{r,s} : \til{\Obs}^{cl}(n) (U) &\to \til{\Obs}^{cl}(n-2) (U) \\
\{-,-\} _{r,s} : \til{\Obs}^{cl}(n) (U) &\to \til{\Obs}^{cl}(n-2) (U)
\end{split}
$$
where $1 \le r \neq s \le n$.
Thus, the spaces $\{ \til{\Obs}^{cl}(n)(U)\}$ are equipped with all the structure maps needed to define a lax $P_0$ algebra. However, in order to show that these spaces actually form a lax $P_0$ algebra, we need to show the following.
''Theorem.'' //For any $n$, the natural map//
$$
\til{\Obs}^{cl}(n)(U) \to \til{\Obs}^{cl}(U)^{\otimes n}
$$
//is a homotopy equivalence. Further, the natural map//
$$
\til{\Obs}^{cl} (U) \to \Obs^{cl}(U)
$$
//is a homotopy equivalence.//
[[Proof | proof of quasi-isomorphism property for lax algebra]]
This result completes the proof of the existence of the desired lax $P_0$ factorization algebra.
Next, we will explain how to construct the $P_0$ structure on the observables of a cotangent field theory. As in the case of a free field theory, we will construct a sub-commutative factorization algebra
$$
\til{\Obs}^{cl} \subset \Obs^{cl}
$$
which is equipped with a $P_0$ structure. For cotangent field theories, however, there is an extra subtlety: the Poisson bracket on $\til{\Obs}^{cl}(U)$ is //not// a bilinear continuous map
$$
\til{\Obs}^{cl}(U) \times \til{\Obs}^{cl}(U) \to \til{\Obs}^{cl}(U).
$$
It is only separately continuous (meaning continuous if either of the variables is kept fixed). This subtlety will also appear when we consider the $P_0$ structure for a general classical field theory. There, we will show that, even with this difficulty, one can use this structure to construct a lax $P_0$ structure on $\Obs^{cl}$.
!!! Cotangent field theories
Thus, let $L$ be an elliptic $L_\infty$ algebra on a manifold $M$. As before, let $\L(U)$ denote the $L_\infty$ algebra associated to an open subset $U \subset M$. Let $\L^!(U)$ denote, as always, the complex of sections of $L^\vee \otimes \op{Dens}_M$. Recall that $L^!(U)$ is a module for $\L(U)$, and that the cotangent field theory associated to $L$ is the the elliptic $L_\infty$ algebra $L \oplus L^![-3]$.
The commutative factorization algebra of classical observables in this context is defined by
$$
\Obs^{cl}(U) = C^\ast ( \L(U)\oplus \L^!(U)[-3] ) .
$$
As we did for a free field theory, we will define a subcomplex $\til{\Obs}^{cl}(U)$ of $\Obs^{cl}(U)$ on which we will define a $P_0$ structure.
This subcomplex will be the ~Chevalley-Eilenberg cochain complex of an enlargment of the $L_\infty$ algebra $\L(U) \oplus \L^!(U)[-3]$. As before, $\br{\L}^!(U)$ will denote the distributional sections of $\L^!(U)$. Note that $\br{\L}^!(U)$ is the dual to the space $\L_c(U)$ of compactly supported sections of $\L(U)$.
The $L_\infty$ action of $\L(U)$ on $\L^!(U)$ is defined by a sequence of polydifferential operators
$$
\L(U)^{\otimes n} \otimes\L^!(U) \to \L^!(U).
$$
Because these maps are polydifferential operators, they extend uniquely to continuous linear maps
$$
\L(U)^{\otimes n} \otimes \br{\L}^!(U) \to \br{\L}^!(U).
$$
These extensions give $\br{\L}^!(U)$ the structure $L_\infty$ module over $\L(U)$.
Of course, this module structure is defined for every open subset, and is compatible with the restriction maps. Thus we see that the sheaf $\br{\L}^!$ of cochain complexes is equipped with an action of the sheaf $\L$ of $L_\infty$ algebras on $M$.
We will let
$$
\til{\Obs}^{cl}(U) = C^\ast ( \L(U) \oplus \br{L}^!(U)[-3] )
$$
be the ~Chevalley-Eilenberg cochains of the semidirect product $L_\infty$ algebra $\L(U) \oplus \br{L}^!(U)[-3]$. It is immediate that $\til{\Obs}^{cl}(U)$ is a commutative factorization algebra, and it follows from the results of the [[appendix|Topological cochain complexes]] that the inclusion
$$
\til{\Obs}^{cl}(U) \to \Obs^{cl}(U)
$$
is a cochain homotopy equivalence.
Thus, to complete our construction, it remains to define a $P_0$ structure on $\til{\Obs}^{cl}(U)$.
Note that
$$
\begin{split}
\til{\Obs}^{cl}(U) &= \what{\Sym}^\ast \left( \L(U)^\vee[-1] \oplus \br{\L}^!(U)^\vee[2] \right) \\
&= \what{\Sym}^\ast \left( \br{\L}^!_c(U)[-1] \oplus \L_c(U) [2] \right)
\end{split}
$$
where these equalities are of graded commutative algebras.
There is a natural pairing on the generators of $\til{\Obs}^{cl}(U)$, arising from the bilinear map
$$
\br{\L}^!_c(U) \times \L_c(U) \to \R.
$$
One needs to be careful at this point, however: this pairing is //not// continuous when we endow $\br{\L}^!_c(U) \times \L_c(U) $ with the product topology. It is only separately continuous, meaning that the pairing $\ip{\phi,\psi}$ is continuous in $\phi$ when $\psi$ is kept fixed, and continuous in $\psi$ when $\phi$ is kept fixed.
A similar issue will arise again when we consider the $P_0$ structure on a factorization algebra in general.
''Lemma.''
// There is a unique separately continuous bilinear map//
$$
\{-,-\} : \til{\Obs}^{cl}(U) \times \til{\Obs}^{cl}(U) \to \til{\Obs}^{cl}(U)
$$
//which satisfies the Leibniz rule, and which on generators $\alpha,\beta \in \br{\L}^!_c(U)[-1] \oplus \L_c(U)[2]$ is defined by//
$$
\{\alpha,\beta\} = \ip{\alpha,\beta}.
$$
''Proof.''
Uniqueness is clear. To see existence, observe that if $\alpha \in \til{\Obs}^{cl}(U)$, then its exterior derivative $\d \alpha$ is an element
$$
\d \alpha \in \til{\Obs}^{cl}(U) \otimes \left(\br{\L}^!_c(U)[-1] \oplus \L_c(U)[2]\right).
$$
Functional analysis results stated in the [[appendix|Overview of nuclear spaces]] imply that
$$
\til{\Obs}^{cl}(U) \otimes \left(\br{\L}^!_c(U)[-1] \oplus \L_c(U)[2]\right) = \Hom \left( \L(U)[1] \oplus \br{L}^!(U)[-2], \til{\Obs}^{cl}(U) \right).
$$
Composing this identification with the natural map
$$
\L_c(U)[1] \oplus \br{L}^!_c(U)[-2] \to \L(U)[1] \oplus \br{L}^!(U)[-2]
$$
gives a map
$$
\til{\Obs}^{cl}(U) \otimes \left(\br{\L}^!_c(U)[-1] \oplus \L_c(U)[2]\right) \to \Hom ( \br{\L}^!_c(U)[-1] \oplus \L_c(U)[2], \til{Obs}^{cl}(U)).
$$
The complex on the right can be identified with the space of continuous derivations of $\til{\Obs}^{cl}(U)$.
Thus, given any $\alpha \in \til{\Obs}^{cl}(U)$, we have constructed a derivation $X_\alpha$ of $\til{\Obs}^{cl}(U)$ (by applying the above sequence of maps to $\d \alpha$).
The Poisson bracket on $\til{\Obs}^{cl}(U)$ is defined by
$$
\{\alpha,\beta\} = X_\alpha (\beta).
$$
It is immediate from the construction that this Poisson bracket is seperately continuous in $\alpha$ and $\beta$, as desired.
$\square$
Thus, we have almost constructed a $P_0$ structrue on $\til{\Obs}^{cl}(U)$: the only difficulty is that the Poisson bracket is not a continuous bilinear map, but is only separately continuous. However, as we will see when we discuss the $P_0$ structure for a general classical field theory, this data is enough to construct a lax $P_0$ algebra structure on $\Obs^{cl}(U)$.
In this page, we will construct a $P_0$ structure on the factorization algebra of observables of a free field theory. More precisely, in each case, we will construct a subcomplex
$$
\til{\Obs}^{cl}(U) \subset \Obs^{cl}(U)
$$
of the complex of classical observables for each open subset $U$, such that
# $\til{\Obs}^{cl}$ forms a sub-commutative facotrization algebra of $\Obs^{cl}$;
# The inclusion $\til{\Obs}^{cl}(U) \subset \Obs^{cl}(U)
$ is a cochain homotopy equivalence, for each $U$;
# $\til{\Obs}^{cl}$ has the structure of $P_0$ factorization algebra.
!!! Free field theories
Recall that a free field theory is a classical field theory associated to an elliptic $L_\infty$ algebra $L$ which is abelian, that is, where all the brackets $\{l_n \mid n \ge 2\}$ vanish.
Thus, let $L$ be an abelian elliptic $L_\infty$ algebra, and let $\L(U)$ be the elliptic complex of sections of $L$ on $U$. To say that $L$ defines a field theory means we have a symmetric isomorphism $L \iso L^! [-3]$.
[[Recall |Classes of sections of a vector bundle]] that we use the notation $\br{\L}(U)$ to denote the space of distributional sections of $L$ on $U$. A lemma of [[Atiyah-Bott |Atiyah-Bott lemma]] shows that the inclusion
$$
\L(U) \into \br{L}(U)
$$
is a homotopy equivalence of topological cochain complexes.
It follows that the natural map
$$
C^\ast ( \br{L}(U) ) \into C^\ast(\L(U))
$$
is a cochain homotopy equivalence. Indeed, because we are dealing with an abelian $L_\infty$ algebra,
$$
\begin{split}
C^\ast ( \L(U)) &= \what{\Sym} ( \L(U)^\vee [-1]) \\
C^\ast ( \br{\L}(U)) &= \what{\Sym} ( \br{\L}(U)^\vee [-1]),
\end{split}
$$
where, as always, the symmetric algebra is defined using the completed tensor product.
Note that
$$
\L(U)^\vee = \br{L}_c^!(U) = \br{L}_c(U)[3].
$$
Thus,
$$
\begin{split}
C^\ast ( \L(U)) &= \what{\Sym} ( \br{\L}_c(U) [2]) \\
C^\ast ( \br{\L}(U)) &= \what{\Sym} ( \L_c(U) [2]).
\end{split}
$$
We can define a Poisson bracket of degree $1$ on $C^\ast(\br{\L}(U)$ as follows. On the generators $\L_c(U)[2]$, it is defined to be the given pairing
$$
\ip{-,-} : \L_c(U) \otimes \L_c(U) \to \R.
$$
This extends uniquely (by the Leibniz rule) to a continuous linear map
$$
C^\ast(\br{\L}(U) \otimes C^\ast(\br{\L}(U) \to C^\ast(\br{\L}(U).
$$
Let us define the modified observables in this complex by
$$
\til{\Obs}^{cl}(U) = C^\ast ( \br{\L}(U)).
$$
We have seen that $\til{\Obs}^{cl}(U)$ is homotopy equivalent to $\Obs^{cl}(U)$, and that $\til{\Obs}^{cl}(U)$ has a $P_0$ structure.
''Lemma.''
// $\Obs^{cl}(U)$ has the structure of a $P_0$ factorization algebra.//
''Proof.'' It remains to verify that, if $U_1,\dots, U_n$ are disjoint open subsets of $M$, each contained in an open subset $W$, then the map
$$
\til{\Obs}^{cl}(U_1) \otimes \dots \otimes \til{\Obs}^{cl}(U_n) \to \til{\Obs}^{cl}(W)
$$
is a map of $P_0$ algebras. This map is automatically a map of commutative algebras; so it suffices to verify that $\alpha \in \til{\Obs}^{cl}(U_i)$, and $\beta \in \til{\Obs}^{cl}(U_j)$, where $i \neq j$, then
$$\{\alpha, \beta\} = 0 \in \til{\Obs}^{cl}(W).$$
This follows immediately from the fact that if $\phi, \psi \in \L_c(W)$ have disjoint support, then
$$
\ip{\phi,\psi} = 0.
$$
$\square$
Our aim here is to demonstrate that applying our procedure to the free scalar field on the real line yields a factorization algebra that is, on any connected interval, quasi-isomorphic to the Weyl algebra $W = \mathbb{C}[[\hbar]]\langle p, q \rangle /(pq - qp - \hbar)$. What this result means is that our procedure recovers the ordinary quantization story in the simplest interesting case. We view this as verification that our procedure gives the "right" answer.
We proceed as follows. First, we pin down the ingredients that allow us to do BV quantization. Then we describe the factorization algebra of classical observables and prove that it is quasi-isomorphic to the appropriate symmetric algebra (namely, the polynomial functions on $T^\ast \mathbb{R}$). We finally construct the factorization algebra of quantum observables and show it is quasi-isomorphic to the Weyl algebra.
!! Ingredients for BV quantization
Our space of fields is
$$
\E = C^\infty(M) \oplus \Omega^1(M)[-1]
$$
where $M$ denotes the real line. Thus, the smooth real-valued functions have cohomological degree 0 and the 1-forms have cohomological degree +1. Our pairing, well-defined only on compactly-supported fields, is the canonical skew-symmetric pairing
$$
\langle (\phi_0, \psi_0 \, dt), \, (\phi_1, \psi_1 \, dt) \rangle = \int_M dt \, \phi_0 \psi_1 - \int_M dt \, \psi_0 \phi_1
$$
where the $\phi$ and $\psi$ denote functions and $t$ denotes a coordinate variable on the real line $M$.
//Remark:// Although the pairing is not well-defined on all fields, we only need it to construct the degree 1 Poisson bracket.
Let $Q$ denote the degree +1, square-zero, skew-symmetric operator
$$
Q: (\phi, \psi \, dt) \mapsto (0, (\triangle \phi) dt).
$$
Our action functional is $S(f) = \langle f, Qf \rangle$, which is just another way of writing $\int_M \phi \triangle \phi dt$. Our gauge-fixing operator $Q^{GF}$ is simply the "shift-down" operator
$$
Q^{GF}: (\phi, \psi \, dt) \mapsto (\psi, 0).
$$
Notice that $D = [Q, Q^{GF}]$ simply applies the Laplacian $\triangle$ to each component.
There are two (families of) operators we need to apply the BV formalism as developed in \cite{webbook}. First, there are the propagators $P(\epsilon, L)$ that allows us to do Feynman diagram computations and run the renormalization group flow from scale $\epsilon$ to scale $L$. Second, we need the "BV Laplacians" $\triangle_L$ which allow us to define the scale $L$ quantum master equation. Both depend on the kernel $K_\ell$, which denotes the kernel of the operator $e^{-\ell D}$. We express this kernel with respect to our skew-symmetric pairing following the convention in \cite{webbook}. The kernel $K_\ell$ is an element of $\E\otimes \E$, and we view it as a section on $M \times M$. We see that
$$
K_\ell(s,t) = \frac{e^{-(s-t)^2/4\ell}}{\sqrt{4\pi \ell}} \left( ds \otimes 1 + 1 \otimes dt \right).
$$
The righthand term has two terms in parantheses meaning "wedge the output by $dx$" and "wedge the input by $dy$," respectively. Note that the kernel has cohomological degree +1.
Given this kernel, the propagator is
$$
P(\epsilon, L) = Q^{GF} \int_\epsilon^L \, K_\ell \, d\ell.
$$
When we plug its associated kernel (which has cohomological degree zero) into Feynman diagrams, it amounts to integrating over the isotropic subspace $\textrm{im} \, Q^{GF}$ of $\E$. The BV Laplacian at scale $\ell$ is
$$
\triangle_\ell = -\partial_{K_\ell}.
$$
This notation simply means "contract with the kernel."
!! The classical observables
Notice that our fields naturally form a sheaf on $M$ where, on an open set $U$, $\E(U)$ denotes $C^\infty(U) \oplus \Omega^1(U)[-1]$. Thus the continuous dual space $\E(U)^\vee$ forms a cosheaf. The dual to $C^\infty(U)$ is the vector space of compactly-supported distributions supported inside $U$. These distributions have cohomological degree 0. Likewise, the dual to $\Omega^1(U)$ is the vector space of compactly-supported currents supported inside $U$. These currents have cohomological degree -1 (so that they pair appropriately with the 1-forms).
The observables
$$
Obs^{cl}(U) := \widehat{\Sym}^\ast (\E(U)^\vee)
$$
form a cosheaf of commutative algebras, with the natural multiplication. Since $Q$ makes $\E$ into a cochain complex, the transpose $Q^\vee$ on $\E^\vee$ extends to a derivation on $Obs^{cl}$. Hence the classical observables form a cosheaf of commutative dgas.
As an example of this differential, consider the following delta function-type observables. Fix a point $x$ in $M$. Let $\phi(x)$ denote the observable which sends the field $(\phi, \psi \, dt)$ to the evaluation of $\phi$ at $x \in M$. Likewise, let $\phi^{(n)}(x)$ denote the evaluation of the n-th derivative $\partial_t^n \phi$ at $x$. Let $\psi(x)$ denote the evaluation of $\psi$ at $x$. Then
$$
Q^\vee (\psi(x)) = \phi^{(2)}(x)
$$
since $Q$ is the Laplacian on the real line. We extend this differential $Q^\vee$ to more complicated monomials by the Leibniz rule.
''Lemma''
The classical observables supported at a point $x$ in $M$ are quasi-isomorphic to the symmetric algebra $\mathbb{C}[q,p]$.
//Proof:// Observe that $\E^\vee_x$ consists of finite $\C$-linear combinations of $\phi(x)$ and its derivatives, $\psi(x)$ and its derivatives. Hence the cohomology of $\E^\vee_x$ is isomorphic to $\mathbb{C}^2$, with natural basis $\phi(x)$ and $\phi^{(1)}(x)$. We identify these with "position" $q$ and "velocity" $p$, respectively. The topology here is irrelevant, so the cohomology of its symmetric algebra $\Sym(\E^\vee)$ is canonically isomorphic to the symmetric algebra of its cohomology $\Sym(H^\cdot (\E^\vee))$. $\square$
In the lemma above, we relied on our knowledge that the distributions at a point are finite linear combinations of the delta function and its derivatives. Standard facts from the theory of distributions tell us that in an open set $U$, the compactly-supported distributions on $U$ are finite linear combinations of continuous functions times derivatives of delta functions. Hence the following lemma should not be surprising.
''Lemma''
For $U$ a connected open interval, $Obs^{cl}(U)$ is quasi-isomorphic to $\mathbb{C}[q,p]$.
//Proof:// This assertion should be reasonable at face value: the cochain complex of fields has cohomology given by the harmonic fields $\{a+bt\}$, so the dual complex should still be the observables on harmonic fields. In this setting of distributions over an interval, however, it is not a trivial assertion, so we give a bare-handed proof.
Fix a point $x$ in $U$. We want to show the inclusion $Obs^{cl}_x \hookrightarrow Obs^{cl}(U)$ is a quasi-isomorphism. Let $C^{-\infty}(U)$ denote the distributions dual to $C^\infty(U)$ (note these are compactly-supported). Let $\Omega^{-1}(U)$ denote the currents dual to $\Omega^1(U)$. We want to show that the cohomology of
$$
\Omega^{-1}(U) \overset{Q^\vee}{\rightarrow} C^{-\infty}(U)
$$
is precisely $\mathbb{C}^2$, where we view $\phi(x)$ and $\phi^{(1)}(x)$ as cocycles forming a basis for the cohomology. We will show, more explicitly, that every observable annihilated by $Q^\vee$ is a sum of the form $a \phi(x) + b \phi^{(1)}(x) + f$, where $a,b$ are complex numbers and $f$ is a $Q^\vee$-coboundary. Note that $Q^\vee$ is injective on $\Omega^{-1}$ (since it sends any term $\psi^{(n)}(y)$ to $\phi^{(n+2)}(y)$), so we can focus our attention on $C^{-\infty}$.
Recall that every such distribution can be expressed (non-uniquely) as a finite sum of derivatives of continuous functions with support inside $U$. Suppose $f$ is a distribution in $C^{-\infty}$ that vanishes on the harmonic fields $\phi(t) = a + bt$. Then we can express $f$ on any field $\phi$ as
$$
\langle f, \phi \rangle =\int_U \alpha_0(t) \phi(t) \, dt + \int_U \alpha_1(t) \phi^{(1)}(t) \, dt + \cdots + \int_U \alpha_n(t) \phi^{(n)}(t) \,dt
$$
for some fixed integer $n$, where the $\alpha_i$ are continuous functions with compact support inside $U$. By hypothesis, we know $\int_U \alpha_0(t) dt = 0$ and $\int_U \alpha_1(t) dt = 0$. We can "integrate away" the term $\alpha_1$ as follows: set
$$
A_1(t) = \int_x^t \alpha_1(s) ds
$$
and observe that
$$
-\int_U A_1(t) \phi^{(2)}(t) \, dt = \int (\partial_t A_1(t)) \phi^{(1)} (t) \, dt = \int_U \alpha_1(t) \phi^{(1)}(t) \, dt.
$$
Thus we might hope to absorb $\alpha_1(t) \phi^{(1)}(t)$ into the "second derivative" piece of $f$. In other words, we change $\alpha_2$ to $\alpha_2 - A_1$ and change $\alpha_1$ to zero. There is only one problem: $A_1$ need not be compactly supported. Here, though, we are rescued by the fact that the total integral of $\alpha_1$ over $U$ is zero. Because $\alpha_1$ has compact support $K \subset U$, it vanishes outside some closed interval $[\lambda,\lambda'] \subset U$. Observe that
$$
\int_\lambda^x \alpha_1(s) \, ds + \int_x^{\lambda'} \alpha_1(s) \, ds = 0 \; \Longrightarrow \; A_1(\lambda) = \int_\lambda^x \alpha_1(s) \, ds = \int_x^{\lambda'} \alpha_1(s) \, ds = A_1(\lambda').
$$
Thus $A_1$ is constant on $U - K'$, taking the same value for $t < \lambda$ and $t > \lambda'$. Denote this constant by $a_1$. Then
$$
\int_U \alpha_1(t) \phi^{(1)}(t) \, dt = \int_U (a_1 - A_1(t)) \phi^{(2)}(t) \, dt,
$$
as integration by parts kills off the constant term $a_1$. Thus, the "first derivative" piece of $f$ lives in the image of $Q^\vee$.
We apply the same process twice to $\alpha_0$ and absorb it into the second derivative as well. Hence $f$ is in the image of $Q^\vee$:
$$
Q^\vee \left( \sum \alpha_i(t) \psi^{(i+2)} \right) = f.
$$
The distributions vanishing on the harmonic fields are $Q^\vee$-coboundaries.$\square$
//Remark:// This lemma leads to a kind of "translation" operator on classical observables. Fix a point $x$ in a connected interval $U$. The most important observables at $x$ are the "position" observable $\phi(x)$ and the "velocity" observable $\phi^{(1)}(x)$. By the lemma, these two observables generate $H^\ast Obs^{cl}(U)$, the observables on $U$. If we consider another point $y$ in $U$,we obtain another natural set of "generators." Let's work out the relationship between the position and velocity observables at $y$, $\phi(y)$ and $\phi^{(1)}(y)$, and the observables that map to $[\phi(x)]$ and $[\phi^{(1)}(x)]$ in $H^\ast Obs^{cl}(U)$. For example, we see that $\phi^{(1)}(x)$ and $\phi^{(1)}(y)$ map to the same cohomology class, and that $\phi(x)$ and $\phi(y) + (x-y) \phi^{(1)}(x)$ map to the same cohomology class. This is just one way of expressing the fact that at time $t = x$, the harmonic field $a + bt$ takes value $a+bx$ and its "velocity" is $b$, and that at time $t=y$, the field takes value $a+by = a + bx + b(y-x)$ and has velocity $b$. We now have a translation map $H^\ast Obs^{cl}_x \rightarrow H^\ast Obs^{cl}_y$ arising from the isomorphism with $H^\ast Obs^{cl}(U)$.
!!! Interpretation
The cochain complex $(\E, Q)$ is the //derived// space of classical states for our field theory. What we mean by this statement is that it is the cochain complex whose cohomology is precisely $\ker \triangle$, so it is a resolution of the space of classical fields (namely the harmonic functions on the real line). Notice that the kernel is two-dimensional and consists of the functions $a+bt$, where $a, b$ are constants that encode "position" at time $t =0$ and "velocity." These functions are recognizable as exactly the trajectories of a free particle on a line, so the simplest observables are certainly polynomials in position and velocity.
!! The quantum observables
Our aim in this section is to show how to work concretely with the ideas and definitions of this wiki. In particular, we want to show that from the factorization algebra of quantum observables we can extract an associative algebra that is a deformation quantization of the symmetric algebra above. Hence, our focus is on defining a "star product" on $Obs^q_x$, the quantum observables supported at a point $x$. Before constructing the product, however, we need to describe the quantum BV complex.
To define the quantum observables, we use the [[local-effective correspondence]]. Let $\ell$ denote the parameter for the length scale $(0,\infty)$. As a graded vector space, the quantum observables $Obs^q$ are the subspace of $Obs^{cl} \otimes_{\mathbb{R}} \mathbb{R}[[\hbar]] \otimes_{\R} \cinfty ((0,\infty))$ given by elements that satisfy the renormalization group equation as a first-order deformation of the free action. Explicitly, $(O[\ell])_{\ell \in (0,\infty)}$ is a quantum observable if
\[
O[L] = W(P(\epsilon,L), \delta O[\epsilon]),
\]
for all $0 < \epsilon < L$, with $\delta$ square-zero of cohomological degree 0. The quantum observables have a differential
\[
\tilde{Q}_\ell = Q^\vee + \hbar \triangle_\ell
\]
that is a deformation of the differential $Q^\vee$ for the classical observables. Recall that the BV differential is scale-dependent, and so we apply $\tilde{Q}_\ell$ to the scale $\ell$ element of a quantum observable. Our differential is particularly simple because our theory is free.
''Lemma''
On a connected open interval $U$, $Obs^q(U)$ is quasi-isomorphic to $\C[q,p][[\hbar]]$ as a vector space.
//Proof:// There is a natural filtration $\{F^k Obs^q\}$ on $Obs^q$ where $F^k$ consists of the observables that vanish mod $\hbar^k$. Consider the associated spectral sequence. Its $E_1$ page consists of $H^\ast Obs^{cl} \otimes \R[[\hbar]]$, where $H^\ast Obs^{cl}$ is concentrated in row 0 (i.e., for the degree arising from the original complex, ignoring the filtration grading). The differential on this page arises from the BV Laplacian $\triangle_\ell$, which lowers this degree by 1, and hence vanishes. Thus the spectral sequence collapses on the $E_1$ page.$\square$
Hence, at the level of $\tilde{Q}$-cohomology, we have the same quasi-isomorphisms as in the classical situation.
Notice that a linear observable is $\tilde{Q}$-closed if and only if $Q^\vee$-closed, because contraction with the heat kernel is a second order differential operator. Hence, we focus our attention on monomials of higher degree.
We would like to make $Obs^q_x$ into an associative algebra, but we cannot multiply two observables with overlapping support because the renormalization group flow leads to divergent integrals. Thanks to the lemmas above, though, we know that for any two points $x$ and $y$ in an open interval $U$, the observables $Obs^{cl}_x$ and $Obs^{cl}_y$ are quasi-isomorphic to $Obs^{cl}(U)$ (see the remark above). By the spectral sequence relating the classical observables to the quantum observables, we see that for the quantum observables $Obs^q_x \hookrightarrow Obs^q(U)$ is a quasi-isomorphism as well. Hence we can construct a quasi-isomorphism $\tau: Obs^q_x \overset{\cong}{\rightarrow} Obs^q_y$. As $Obs^q$ is a factorization algebra, we have a "multiplication" map $m_{xy}: Obs^q_x \otimes Obs^q_y \rightarrow Obs^q(U)$. We define the star product as the composition
$$
\star: Obs^q_x \otimes Obs^q_x \overset{1\otimes \tau}{\longrightarrow} Obs^q_x \otimes Obs^q_y \overset{m}{\longrightarrow} Obs^q(U).
$$
This product on the cochain complex induces a unique, well-defined star product on the cohomology $H^\ast(Obs^q_x, \tilde{Q})$. We will see that this product, at the level of cohomology, is independent of $y$, for fixed, but arbitrary, $x$.
We begin with some preliminary computations. Consider the observables $q = \phi(x)$ and $p = \phi^{(1)}(x)$ in $Obs_x$. Then
$$
\tau (q) = \phi(y) + (x-y) \phi^{(1)}(y)
$$
and
$$
\tau (p) = \phi^{(1)}(y).
$$
These classical observables can be identified with obvious quantum observables (since the RG flow does nothing to them). To compute the product $m(q, \tau(p)) = a \cdot \tau (p)$ as quantum observables, we need to identify a quantum observable whose scale 0 limit, under the RG flow, corresponds to the classical product.
In this case, we are lucky, as we can simply apply the renormalization group flow to $q \cdot \tau( p)$ starting at scale 0. This flow uses the kernel of the propagator. Write the heat kernel as $K_\ell(s,t)$, where $(s,t) \in M \times M$. We find
$$
(q \cdot \tau (p))[L] := \phi(x) \phi^{(1)}(y) - \hbar\int_0^L \, \partial_t \Big|_y K_\ell(x,t)\, d\ell.
$$
Likewise, we can compute $p \cdot \tau(q)$:
$$
(p \cdot \tau (q))[L] := \phi^{(1)}(x) \phi(y) - \hbar\int_0^L \partial_s \Big|_x K_\ell (s,y) \, d\ell + \hbar (x-y) \int_0^L \partial_s \Big|_x \partial_t\Big|_y K_\ell (s,t) \, d\ell.
$$
It will be useful for our computations below to observe that
$$
\partial_s K_\ell = -\partial_t K_\ell \mbox{ and } \partial_\ell K_\ell = \partial_s^2 K_\ell = -\partial_s \partial_t K_\ell
$$
by direct computation. The second equation is simply the statement that the heat kernel satisfies the heat equation.
''Lemma''
These products $q \cdot \tau(p)$ and $p \cdot \tau(q)$ are $\tilde{Q}$-closed.
//Proof:// These are $Q$-closed, as seen above. So it suffices to check that the BV Laplacian annihilates them. Note that the heat kernel $K_L$ has cohomological degree +1. Thus contraction with the heat kernel kills observables of cohomological degree zero, such as the products we're considering.$\square$
''Lemma''
At the level of $\tilde{Q}$-cohomology, the commutator bracket with respect to the star product is $[p,q] = -\hbar$.
//Proof:// We compute the bracket $B = (p \cdot \tau(q) - q \cdot \tau(p))[L]$. Observe that the classical contributions cancel (i.e., the $\hbar^0$ components cancel as the classical product commutes). The $\hbar$ component of $B$ is
$$
2 \int_0^L \partial_t \Big|_y K_\ell(x,t) \, d\ell + (x-y) \int_0^L \partial_s \Big|_x \partial_t \Big|_y K_\ell(s,t) \, d\ell
$$
$$
= 2 \int_0^L \partial_t \Big|_y K_\ell(x,t) \, d\ell - (x-y) K_L(x,y),
$$
using the heat equation relation and the fact that $K_0(x,y) = 0$ for $x \neq y$.
Our approach is to find an element $A$ whose image $\tilde{Q}(A)$ differs from the product above by a constant. Then we will know our commutator is a constant at the level of $\tilde{Q}$-cohomology.
Let
$$
A := -\phi(y) \int_x^y \psi(a) \, da+ \phi^{(1)}(y) \int_x^y \int_x^a \psi(b) \, db \, da.
$$
Observe that
$$
Q(A) = -\phi(y) \int_x^y \phi^{(2)}(a) \, da+ \phi^{(1)}(y) \int_x^y \int_x^a \phi^{(2)}(b) \, db \, da
$$
$$
= -\phi(y) (\phi^{(1)}(y) - \phi^{(1)}(x)) + \phi^{(1)}(y) (\phi(y) - \phi(x) - (y-x) \phi^{(1)}(x))
$$
$$
= \phi^{(1)}(x) \phi(y) - \phi(x) \phi^{(1)}(y) + (x-y) \phi^{(1)}(x) \phi^{(1)}(y).
$$
Hence the classical contributions agree with those from $B$ and thus vanish. The BV Laplacian contributes the $\hbar$ term:
$$
-\int_x^y K_L(a,y) \, da+ \int_x^y \int_x^a \partial_t \Big|_y K_L(b,t) \, db \, da
$$
$$
=-\int_x^y K_L(a,y) \, da - \int_x^y \int_x^a \partial_b K_L(b,y) \, db \, da
$$
using integration by parts, and now
$$
= -\int_x^y K_L(a,y) \, da - \int_x^y K_L(a,y) \, da - (y-x) K_L(x,y)
$$
$$
= -2 \int_x^y K_L(a,y) \, da + (x-y) K_L(x,y).
$$
This $\hbar$ term combines nicely with $B$'s term.
We compute, substituting $s$ for $a$, that
$$
B + \tilde{Q}(A) = 2\int_0^L \partial_t \Big|_y K_\ell(x,t) \, d\ell -2 \int_x^y K_L(s,y) \, ds .
$$
To see that it is independent of $L$, we take the derivative
$$
\partial_L (B + \tilde{Q}(A)) = 2\left( \partial_y K_\ell(x,y) - \int_x^y \partial_L K_L(s,y) \, ds \right).
$$
and using the heat equation relation we see
$$
= 2\left( \partial_y K_\ell(x,y) - \int_x^y \partial^2_s K_L(s,y) \, ds \right)
$$
$$
= 2\left( \partial_y K_\ell(x,y) - \partial_s K_L(y,y) + \partial_s K_L(x,y)\right).
$$
Since $\partial_s K_L = - \partial_t K_L$, we see that the terms involving $x$ and $y$ cancel. The term depending solely on $y$ vanishes because the heat kernel is constant along the diagonal. Hence $B + \tilde{Q}(A)$ is constant as a function of $L$.
We can compute the constant as follows. Since the sum $B + \tilde{Q}(A)$ is constant, we evaluate the associated integrals in the limit as $L$ goes to zero. Thus we need to compute
$$
\lim_{L \rightarrow 0} -2 \int_x^y K_L(a,y) \, da.
$$
As $L$ becomes small, the heat kernel approaches a delta function supported at $(y,y)$, so the limit is $-1$. To make this completely clear, observe that the heat kernel is symmetric across the diagonal. Hence
$$
-2 \int_x^y K_L(a,y) \, da = - \int_x^{y+x} K_L(a,y) \, da.
$$
Thus as $L$ goes to zero, this becomes integration of the delta function supported at $y$. The constant is thus $-\hbar$ and so the commutator is cohomologous to $-\hbar$.$\square$
This lemma shows that we get the defining relation of the Weyl algebra. In fact, this relation determines the star product.
''Theorem''
The star product on $H^\ast(Obs^q_x, \tilde{Q})$ makes the cohomology into an associative algebra isomorphic to the Weyl algebra.
//Proof://
We have shown that on every connected open interval $U$, $H^\ast Obs^q(U)$ is quasi-isomorphic to $\C [[\hbar]] [ q,p]$ as a vector space. We have also shown that given two connected open intervals $U_1, U_2$ contained inside a connected interval $V$, the map
\[
H^\ast Obs^q(U_1) \otimes H^\ast Obs^q(U_2) \rightarrow H^\ast Obs^q(V)
\]
given by the factorization algebra structure is the same as the product in the Weyl algebra. Since $H^\ast Obs^q$ is a factorization algebra in vector spaces, we have the associativity relation. To summarize, the Weyl algebra gives a factorization algebra on the real line and it is isomorphic to $H^\ast Obs^q$.$\square$
In this page, we explain how prefactorization algebras and factorization algebras form categories. In fact, they naturally form multicategories (or colored operads). We also explain how these multicategories are enriched in simplicial sets when the (pre)factorization algebras take values in cochain complexes.
!! Morphisms and the category structure
''Definition''
A morphism of prefactorization algebras $\phi: F \to G$ consists of a map $\phi_U: F(U) \to G(U)$ for each open $U \subset M$, compatible with the structure maps. That is, for any open $V$ and any finite collection $U_1, \ldots, U_k$ of pairwise disjoint open sets, each contained in $V$, the following diagram commutes:
\[
\begin{matrix}
F(U_1) \otimes \cdots \otimes F(U_k) & \overset{\phi_{U_1} \otimes \cdots \otimes \phi_{U_k} }{\longrightarrow} & G(U_1) \otimes \cdots \otimes G(U_k) \\
\downarrow & & \downarrow \\
F(V) & \overset{\phi_V}{ \longrightarrow } & G(V)
\end{matrix}
\]
Likewise, all the obvious associativity relations are respected.
//Remark:// When our prefactorization algebras take values in cochain complexes, we require the $\phi_U$ to be cochain maps, i.e., they each have degree 0 and commute with the differentials.
''Definition''
On a space $X$, we denote the category of prefactorization algebras on $X$ taking values in the symmetric monoidal category $\mathcal{C}$ by $PreFA(X, \mathcal{C})$. The category of factorization algebras, $FA(X, \mathcal{C})$, is the full subcategory whose objects are the factorization algebras.
!! The multicategory structure
There is a natural tensor product on $PreFA(X,\mathcal{C})$, as follows. Let $F, G$ be prefactorization algebras. We define $F \otimes G$ by
\[
F\otimes G(U) := F(U) \otimes G(U),
\]
and we simply define the structure maps as the tensor product of the structure maps. For instance, if $U \subset V$, then the structure map is
\[
F(U \subset V) \otimes G(U \subset V): F \otimes G(U) = F(U) \otimes G(U) \to F(V) \otimes G(V) = F \otimes G(V).
\]
''Definition''
Let $PreFA_{mc}(X, \mathcal{C})$ denote the multicategory arising from the symmetric monoidal product on $PreFA(X, \mathcal{C})$. That is,
\[
PreFA_{mc}(F_1, \cdots, F_n ; G) := PreFA( F_1 \otimes \cdots \otimes F_n, G).
\]
Factorization algebras inherit this multicategory structure.
!! Enrichment over simplicial sets
Recall that cochain complexes are enriched over simplicial sets as follows. For $K$ a simplicial set,
\[
sSets(K, Maps(A,B)) = Comp_0( \Omega^*(K) \otimes A, B),
\]
where $Comp_0(A,B)$ denotes the cochain maps, i.e., degree zero maps that commute with the differentials, and $\Omega^* (K)$ denotes the de Rham complex on the geometric realization of $K$. In particular, the n-simplices of $Maps(A,B)$ are precisely $Comp_0(\Omega^*(\Delta^n) \otimes A, B)$. Let $Comp$ denote the category whose objects are cochain complexes and whose morphisms from $A$ to $B$ is the mapping space $Maps(A,B)$. This category is enriched over simplicial sets.
We use this same method to enrich prefactorization algebras over simplicial sets. Given prefactorization algebras $F, G$ taking values in cochain complexes, define $Maps(F,G)$ as follows. An $n$-simplex $\phi$ in $Maps(F,G)$ consists of a map $\phi_U \in Comp_0(\Omega^*(\Delta^n) \otimes F(U), G(U))$ for each open $U \subset $X$, compatible with the structure maps.
<<getTiddlerPassword>>
Let $FactAlg(M, \sC^\otimes)$ denote the category of factorization algebras on the space $M$ taking value in the symmetric monoidal category $\sC^\otimes = (\sC, \otimes)$. Given an open subset $U \subset M$, there is an obvious restriction map $FactAlg(M, \sC^\otimes) \rightarrow FactAlg(U,\sC^\otimes)$ that simply forgets about the behavior of a factorization algebra $F$ on open sets $V \subset M$ not contained in $U$. In other words, we have a functor
\[
FactAlg( - , \sC^\otimes): Open_M^{op} \rightarrow Cat.
\]
We thus have a presheaf. A natural question is whether this functor is in fact a sheaf.
We consider a related, simpler question. Let $\text{Iso } FactAlg(M, \sC^\otimes)$ denote the category whose objects are factorization algebras on $M$ and whose morphisms are isomorphisms of such factorization algebras. This category is the underlying groupoid of $FactAlg(M, \sC^\otimes)$.
''Proposition''
The functor
\[
\text{Iso } FactAlg( - , \sC^\otimes): Open_M^{op} \rightarrow Gpds.
\]
is a sheaf.
//Proof:// We need to verify descent. Let $U \subset M$ be an open and let $\mathfrak U = \{U_\alpha \}$ be an open cover of $U$. Let $CN(\mathfrak U)$ denote the \v{C}ech nerve of the cover. We compute the limit over the nerve:
\[
\sL = \lim_{CN(\mathfrak U)} \text{Iso } FactAlg(-,\sC^\otimes).
\]
Since $\text{Iso } FactAlg(U,\sC^\otimes)$ maps coherently to every groupoid in the diagram over $CN(U)$, we clearly have a map
\[
i: \text{Iso } FactAlg(U,\sC^\otimes) \rightarrow \sL.
\]
We wish to show this is an equivalence.
Here we use the locality axiom. A factorization algebra on $U$ is determined by its behavior on the open cover. In particular, we can find a factorizing basis for $U$ subordinate to the cover $\mathfrak U$. Using this basis, we can compute the values of the factorization algebra on any open subset of $U$. In other words, up to automorphism, there is precisely one factorization algebra on $U$ arising from a coherent collection of factorization algebras over the \v{C}ech nerve. The case of morphisms is likewise. A morphism of factorization algebras on $U$ is determined by the morphism over a factorizing basis. We now make these assertions more precise.
By construction, there is a functor $r_V: \sL \rightarrow \text{Iso } FactAlg(V, \sC^\otimes)$ for each open $V \in CN(\mathfrak U)$.
We start with objects. Let $L$ denote an object of $\sL$. We will show there is a factorization algebra $F_L$ on $U$, unique up to isomorphism, such that $i(F_L) = L$. Construct $F_L$ as follows. For each open $V \in CN(\mathfrak U)$, there is a factorization algebra $L_V = r_V(L)$ on $V$, so set $F_L \big|_V = L_V$. We need to determine $F_L$ on open sets that are not contained in some element of the cover $\mathfrak U$. Pick a basis $\mathfrak B = \{B_\alpha\}$ of open sets for $U$ that is factorizing and subordinate to the cover $\mathfrak U$. For every open $W \subset U$, let $\mathfrak{B}_W \subset \mathfrak B$ consist of all opens $B_\alpha \in \mathfrak B$ contained in $W$. It gives a factorizing basis of $W$. We define $F_L(W)$ to be the colimit using the cover $\mathfrak{B}_W$, as specified by the gluing axiom. Note that if $ W \subset W' \subset U$, then $\mathfrak{B}_W \subset \mathfrak{B}_{W'}$, so by construction we obtain a map $F_L(W) \rightarrow F_L(W')$. Thus we have a prefactorization algebra $F_L$, and it remains to show that $F_L$ is a factorization algebra.
Pick any open $W \subset U$ and any factorizing cover $\mathfrak W = \{W_\alpha\}$ of $W$. We need to show that $F_L(W)$ agrees with the colimit computed using the factorizing cover $\mathfrak W$. Observe that for each $W_\alpha$, $F_L(W_\alpha)$ is computed using the factorizing cover $\mathfrak{B}_{W_\alpha}$. Hence the colimit over $\mathfrak W$ is equivalent to the colimit over the $\bigcup_\alpha \mathfrak{B}_{W_\alpha}$. We need to show this colimit agrees with the colimit over $\mathfrak{B}_W$.
Observe that $\bigcup_\alpha \mathfrak{B}_{W_\alpha}$ is a factorizing basis of $W$. It is a basis because given any open $V \subset W$,
\[
V = \bigcup_\alpha (V \cap W_\alpha) = \bigcup_\alpha \bigcup_\beta B_{\alpha, \beta},
\]
where $B_{\alpha,\beta} \in \mathfrak{B}_{W_\alpha}$ and $V \cap W_\alpha = \bigcup_\beta B_{\alpha,\beta}$. In words, since $\mathfrak W$ is a cover and we have a factorizing basis for each open in that cover, we can construct any open inside $W$. And this basis is factorizing because $\mathfrak W$ is.
Since any $B \in \mathfrak{B}_W$ is contained in some $V \in CN(\mathfrak U)$, $F_L$ restricted to $B$ is a factorization algebra and is thus determined by its behavior on a factorizing basis, notably $\bigcup_\alpha \mathfrak{B}_{W_\alpha}$. This insures that the colimit over $\mathfrak{B}_W$ agrees with the colimit over $\mathfrak W$. Moreover, we see that $F_L \big|_V$ is isomorphic to $r_V(L)$.
The argument for morphisms is similar. A morphism of factorization algebras is determined by its behavior on a factorizing basis. Thus, given a morphism $\phi: L \rightarrow L'$ in $\sL$, we obtain a morphism of factorization algebras $F(\phi): F_L \rightarrow F_{L'}$ that is unique up to isomorphism. Moreover, $i(F(\phi)) \cong \phi$ since they agree under restriction to any open $V$ in the \v{C}ech nerve.\square
Before we discuss the ~Batalin-Vilkovisky formalism for classical field theory, we will discuss a finite-dimensional toy model (which we can think of as a $0$-dimensional classical field theory). Our model for the space of fields is a finite-dimensional smooth manifold manifold $M$. The "action functional" is given by a smooth function $S \in \cinfty(M)$. Classical field theory is concerned with solutions to the equations of motion. In our setting, the equations of motion are given by the subspace $\op{Crit}(S) \subset M$. Our toy model will not change if $M$ is a smooth algebraic variety or a complex manifold, or indeed a smooth formal scheme. Thus we will write $\Oo(M)$ to indicate whatever class of functions (smooth, polynomial, holomorphic, power series) we are considering on $M$.
If $S$ is not a nice function, then this critical set can by highly singular. The classical ~Batalin-Vilkovisky formalism tells us to take, instead the //derived// critical locus of $S$. (Of course, this is exactly what a derived algebraic geometer ([[Lur09]], [[Toe06]]) would tell us to do as well).
The critical locus of $S$ is the intersection of the graph
$$\Gamma( \d S) \subset T^\ast M$$
with the zero-section of the cotangent bundle of $M$. Algebraically, this means that we can write the algebra $\Oo(\op{Crit}(S))$ of functions on $\op{Crit}(S)$ as a tensor product
$$
\Oo(\op{Crit}(S)) = \Oo( \Gamma ( \d S) ) \otimes_{\Oo( T^\ast M ) } \Oo (M) .
$$
Derived algebra geometry tells us that the derived critical locus is obtained by replacing this tensor product with a derived tensor product. Thus, the derived critical locus of $S$ (which we denote $\op{Crit}^h(S)$ is an object such that
$$
\Oo(\op{Crit}^h(S)) = \Oo( \Gamma ( \d S) ) \otimes^{\mbb{L}}_{\Oo( T^\ast M ) } \Oo (M) .
$$
In derived algebraic geometry, as in ordinary geometry, spaces are determined by their algebras of functions. In derived geometry, however, one allows differential-graded algebras as algebras of functions (normally one restricts attention to differential-graded algebras concentrated in non-positive cohomological degrees).
We will take this derived tensor product as a definition of $\Oo(\op{Crit}^h(S))$.
!!! An explicit model
It is convenient to consider an explicit model for the derived tensor product. By taking a standard Koszul resolution of $\Oo( M)$ as a module over $\Oo(T^\ast M)$, one sees that $\Oo(\op{Crit}^h(S))$ can be realized as the complex
$$
\Oo(\op{Crit}^h(S)) \simeq \dots \xto{\vee \d S} \Gamma (M, \wedge^2 T M ) \xto{\vee \d S} \Gamma (M, TM ) \xto{\vee \d S} \Oo(M).
$$
In other words, we can identifty $\Oo(\op{Crit}^h (S))$ with functions on the graded manifold $T^\ast[-1] M$, equipped with the differential given by contracting with $\d S$.
Note that
$$
\Oo( T^\ast[-1] M ) = \Gamma (M , \wedge^\ast TM )
$$
has a Poisson bracket of cohomological degree $1$, called the ~Schouten-Nijenhuis bracket. This Poisson bracket is characterized by the fact that if $f, g \in \Oo(M)$ and $X, Y \in \Gamma (M, T M)$, then
$$
\begin{split}
\{X,Y\} &= [X, Y] \\
\{X, f\} &= X f \\
\{f, g \} & = 0
\end{split}
$$
(the Poisson bracket between other elements of $\Oo(T^\ast[-1] M)$ is inferred from the Leibniz rule).
The differential on $\Oo(T^\ast [-1] M)$ corresponding to that on $\Oo(\op{Crit}^h(S))$ is given by
$$
\d \phi = \{S, \phi\}
$$
for $\phi \in \Oo( T^\ast [-1] M)$.
The derived critical locus of any function is a dg manifold equipped with a symplectic form of cohomological degree $-1$. In the ~Batalin-Vilkovisky formalism, the space of fields always has such a symplectic structure. However, one does not require that the space of fields arises as the derived critical locus of a function.
We would like to consider classical field theories in the BV formalism. For us, such a classical field theory will be specified by an elliptic moduli problem equipped with a symplectic form of cohomological degree $-1$.
We defined the notion of formal elliptic moduli problem on a manifold $M$ using the language of $L_\infty$ algebras. Thus, in order to give the definition of a classical field theory, we need to understand the following question: what extra structure on an $L_\infty$ algebra $\g$ endows the corresponding formal moduli problem with a symplectic form?
The answer to this question was given by Kontsevich ([[Kon93]]). Given a pointed formal moduli problem $\mc{M}$, the associated $L_\infty$ algebra $\g_{\mc{M}}$ has the property that
$$
\g_{\mc{M}} = T_p \mc{M} [-1].
$$
Further, we can identify geometric objects on $\mc{M}$ in terms of $\g_{\mc{M}}$ as follows.
| $C^\ast(\g_{\mc{M}})$ | The algebra $\Oo(\mc{M})$ of functions on $\mc{M}$ |
|$\g_{\mc{M}}$-modules $V$ | $\Oo_{\mc{M}}$-modules |
| $C^\ast(\g_{\mc{M}}, V )$ | the $\Oo_{\mc{M}}$ module |
| The $\g_{\mc{M}}$-module $g_{\mc{M}}[1]$ | $T \mc{M}$ |
Following this logic, we see that the complex of two-forms on $\mc{M}$ can be identified with $C^\ast(\g_{\mc{M}}, \wedge^2 ( \g_{\mc{M}}^\vee [-1] ) )$.
However, on a symplectic formal manifold, one can always choose Darboux coordinates. Changes of coordinates on $\mc{M}$ correspond to $L_\infty$ isomorphisms on $\g_{\mc{M}}$. In Darboux coordinates, the symplectic form has constant coefficients, and thus can be viewed as a $\g_{\mc{M}}$-invariant element of $\wedge^2 ( g_{\mc{M}}^\vee [-1])$.
Note that the usual Koszul rules of signs imply that
$$
\wedge^2 ( g_{\mc{M}}^\vee [-1]) = \Sym^2 (\g^\vee_{\mc{M}} ) [-2].
$$
To give a $\g_{\mc{M}}$-invariant element of $\Sym^2 (\g^\vee_{\mc{M}} )$ is the same as to give an invariant symmetric bilinear form on $\g_{\mc{M}}$.
Thus, we arrive at the following principle:
<<<
To give a formal pointed derived moduli problem with a symplectic form of cohomological degree $k$ is the same as to give an $L_\infty$ algebra with an invariant and non-degenerate pairing of cohomological degree $k-2$.
<<<
We will define a classical field theory to be an elliptic $L_\infty$ algebra equipped with a non-degenerate invariant pairing of cohomological degree $-3$. Let us first define what it means to have an invariant pairing on an elliptic $L_\infty$ algebra.
''Definition''
//Let $M$ be a manifold, and let $E$ be an elliptic $L_\infty$ algebra on $M$. An invariant pairing on $E$ of cohomological degree $k$ is a symmetric vector bundle map//
$$
\ip{-,-}_E : E \otimes E \op{Dens}(M) [k]
$$
//satisfying some additional conditions://
* //Non-degeneracy: we require that this pairing induces a vector bundle isomorphism//
$$
E \to E^\vee \otimes \op{Dens}(M) [-3].
$$
*// Invariance: let $\E_c$ denotes the space of compactly supported sections of $E$. The pairing on $E$ induces an inner product on $\E_c$, defined by//
$$
\begin{split}
\ip{-,-} : \E_c \otimes \E_c &\to \R \\
\alpha \otimes \beta & \to \int_M \ip{\alpha,\beta}.
\end{split}
$$
//We require that this is an invariant pairing on the $L_\infty$ algebra $\E_c$. //
Recall that a symmetric pairing on an $L_\infty$ algebra $\g$ is called invariant if, for all $n$, the linear map
$$
\begin{split}
\g^{\otimes n+1} &\to \R \\
\alpha_1 \otimes \dots \otimes \alpha_{n+1} & \mapsto \ip{l_n(\alpha_1,\dots,\alpha_n), \alpha_{n+1}}
\end{split}
$$
is graded anti-symmetric in the $\alpha_i$.
''Definition''
//A formal pointed elliptic moduli problem on with a symplectic form of cohomological degree $k$ on a manifold $M$ is an elliptic $L_\infty$ algebra on $M$ with an invariant pairing of cohomological degree $k-2$.//
''Definition''
//A (perturbative) classical field theory on $M$ in the BV formalism is a formal pointed elliptic moduli problem on $M$ with a symplectic form of cohomological degree $-1$. //
The critical locus of a function $f$ is, of course, the zero locus of the one-form $\d f$. We are interested in constructing the derived critical locus of a local functional $S \in \Ool( B \L)$ on the formal moduli problem associated to a local $L_\infty$ algebra on a manifold $M$. Thus, we need to understand what kind of object the exterior derivative $\d S$ of such an $S$ is.
If $\g$ is an $L_\infty$ algebra, then we should think of $C^\ast_{red}(\g)$ as the algebra of functions on the formal moduli problem $B \g$ associated to $\g$, which vanish at the base point. Similarly, $C^\ast(\g, \g^\vee[-1])$ should be the thought of as the space of one-forms on $B \g$. The exterior derivative is thus a map
$$
C^\ast_{red}(\g) \to C^\ast(\g, \g^\vee[-1] ) .
$$
We will define a similar the exterior deriviative for a local Lie algebra $L$ on $M$. The analog of $\g^\vee$ is the $L$-module $L^!$. Thus, our exterior derivative will be a map
$$
\Ool(B \L ) \to C^\ast_{loc} ( \L , \L^! [-1] ) .
$$
Recall that $\Ool ( B \L)$ is a subcomplex of $C^\ast_{red}( \L_c(M))$. The exterior derivative for the $L_\infty$ algebra $\L_c(M)$ is a map
$$
\d : C^\ast_{red}( \L_c(M)) \to C^\ast(\L_c(M), \L_c(M)^\vee[-1] ).
$$
Note that the dual $\L_c(M)^\vee$ of $\L_c(M)$ is the space $\br{\L}^!(M)$ of distributional sections of the bundle $L^!$ on $M$. Thus, the exterior derivative can be viewed as a map
$$
\d : C^\ast_{red}( \L_c(M)) \to C^\ast(\L_c(M), \br{\L}^!(M)[-1] ).
$$
Note that
$$
C^\ast_{loc}( \L, \L^![-1] ) \subset C^\ast ( \L_c(M), \L^!(M) ) \subset C^\ast ( \L_c(M), \br{\L}^!(M ) ).
$$
''Lemma.''
//The exterior derivative takes the subspace $\Ool (B \L )$ of $C^\ast_{red}( \L_c(M))$ to the subspace $C^\ast_{loc}( \L, \L^![-1] ) $ of $C^\ast( \L_c(M), \br{\L}^!(M))$.//
''Proof.'' The content of this lemma is the equivalent to the familiar statement that the Euler-Lagrange equations associated to a local action functional are differential equations. We will give a formal proof, but the reader will see that all that is used is integration by parts.
Any functional
$$
F \in \Ool ( B \L )
$$
can be written as a sum $F = \sum F_n$ where
$$
F_n \in \op{Dens}_M \otimes_{D_M} \Hom_{\cinfty_M} \left( J(L)^{\otimes n}, \cinfty_M \right)_{S_n}.
$$
Any such $F_n$ can be written as a finite sum
$$
F_n = \sum_i \omega D^{i}_1 \dots D^i_n
$$
where $\omega$ is a section of $\op{Dens}_M$ and $D_j^i$ are differential operators from $\L \to \cinfty_M$.
If we view $F \in \Oo(\L_c(M))$, then the $n^{th}$ Taylor component of $F$ is the linear map
$$
\L_c(M)^{\otimes n} \to \R
$$
defined by
$$
\phi_1 \otimes \dots \otimes \phi_n \to \sum_i \int_M \omega (D^i_1 \phi_1) \dots (D^i_n \phi_n) .
$$
Thus, the $(n-1)^{th}$ Taylor component of $\d F$ is given by the linear map
$$
\begin{split}
\d F_n : \L_c(M)^{\otimes n-1} &\to \br{L}^!(M) = \L_c(M)^\vee \\
\phi_1 \otimes \dots \otimes \phi_{n-1} & \sum_i \mapsto \omega (D^i_1 \phi_1) \dots (D^i_{n-1} \phi_{n-1})D_n^i ( - ) + \text{symmetric terms}
\end{split}
$$
where the right hand side is viewed as a linear map from $\L_c(M)$ to $\R$. Now, by integration by parts, we see that
$$
(\d F_n)(\phi_1,\dots,\phi_{n-1} )
$$
is in the subspace $\L^{!}(M) \subset \br{L}^!(M)$ of smooth sections of the bundle $L^!(M)$, inside the space of distributional sections.
It is clear from the explicit expressions that the map
$$
\d F_n : \L_c(M)^{\otimes n - 1} \to \L^!(M)
$$
is a polydifferential operator, and so defines an element of $C^\ast_{loc}(\L, \L^![-1])$ as desired.
$\square$
We have [[seen | the prefactorization algebra of observables]] how to define a prefactorization algebra of local observables for our quantum field theory. In this page we will show that this prefactorization algebra is in fact a factorization algebra. In the course of the proof, we show that modulo $\hbar$, this factorization algebra is isomorphic to $\Obs^{cl}$.
''Theorem.''
* //The prefactorization algebra $\ObsStrict$ of quantum observables is, in fact, a factorization algebra.//
* //Further, there is an isomorphism//
$$
\ObsStrict \otimes_{\mathbb{R}[[\hbar]]} \mathbb{R} \cong \Obs^{cl}
$$
//between the $\operatorname{mod}\, \hbar$ reduction of the factorization algebra of quantum observables, and the factorization algebra of classical observables.//
!!! Proof of the theorem
This theorem will be a corollary of a more technical proposition.
''Proposition''
//For any open subset $U \subset M$, let us filter $\ObsStrict(U)$ by saying that $F^k \ObsStrict(U)$ is the sub $\mathbb{R}[[\hbar]]$-module consisting of those observables which are zero modulo $\hbar^k$. Note that this is a filtration by sub prefactorization algebras over the ring $\mathbb{R}[[\hbar]]$.//
//Then, there is an isomorphism of prefactorization algebras//
$$
\operatorname{Gr} \ObsStrict \simeq \Obs^{cl} \otimes_{\mathbb{R}} \mathbb{R}[[\hbar]].
$$
This isomorphism makes $\operatorname{Gr} \ObsStrict$ into a factorization algebra.
''Proof of the theorem, assuming the proposition.''
We need to show that for every open $U$ and for every factorizing cover $\mf{U}$, we have a quasi-isomorphism
$$
\check{C} ( \mf{U}, \Obs) \simeq \Obs(U).
$$
The basic idea is that the filtration induces a spectral sequence for both $\check{C} ( \mf{U}, \Obs)$ and $\Obs(U)$, and we will show that there is a map of spectral sequences which is an isomorphism on the first page. Note that our filtration on $\Obs$ is complete, so these spectral sequence converge.
The filtration yields a spectral sequence converging to $\check{C} ( \mf{U}, \Obs)$ whose first page is $\check{C} (\mf{U}, \operatorname{Gr} \ObsStrict(U))$. Likewise, the filtration yields a spectral sequence converging to $\Obs(U)$ whose first page is$\operatorname{Gr} \ObsStrict(U)$. By the proposition, we know there is a quasi-isomorphism between the first pages, as $\operatorname{Gr} \ObsStrict$ is a factorization algebra.
$\square$
''Proof of the proposition.''
The first step in the proof of the proposition is the following lemma.
''Lemma.''
//Let $\ObsStrict^{(0)}$ denote the prefactorization algebra of observables which are only defined modulo $\hbar$. Then, there is an isomorphism of prefactorization algebras//
$$
\ObsStrict^{(0)} \simeq \Obs^{cl}
$$
//of differential graded prefactorization algebras.//
''Proof of lemma.''
Suppose that $O \in \Obs^{cl}(U)$ is a classical observable. Thus, $O$ is an element of the cochain complex $\mathcal{O}(\mathcal{E}\mid U)$ of functionals on the space of fields on $U$. From this, we need to produce an element of $\ObsStrict^{(0)} $. An element of $\ObsStrict^{(0)}$ is a collection of functionals $O[\Phi] \in \mathcal{O}(\mathcal{E})$, one for every parametrix $\Phi$, satisfying a classical version of the renormalization group equation, and an axiom saying that $O[\Phi]$ is supported on $U$ for sufficiently small $\Phi$.
Given an element
$$
O \in \Obs^{cl}(U) = \mathcal{O}( \mathcal{E} \mid_U) ,
$$
we define an element
$$
\{ O[\Phi] \} \in \ObsStrict^{(0)}
$$
by the formula
$$
\delta O[\Phi] = \operatorname{lim}_{P' \to 0} W ( P (\Phi) - P(\Psi), I[\Psi] + \delta O[\Psi] ) - I[\Phi] \text{ modulo } \hbar.
$$
The Feynman diagram expansion of the right hand side only involves trees, since we are working modulo $\hbar$. The fact that we are only using trees implies the limit exists.
The map
$$
\Obs^{cl}(U) \to \ObsStrict^{(0)}(U)
$$
is easily seen to be a map of cochain complexes, compatible with the structure of prefactorization algebra present on both sides. (It is a variation on the argument in section 11, chapter 5 of \cite{webbook}, about the scale 0 limit of a deformation of $I$ modulo $\hbar$.)
A simple inductive argument on the degree shows that this map is an isomorphism.
$\square$
The next (and most difficult) step in the proof of the proposition is the following lemma.
Let $\ObsStrict^{(k)}$ denote the prefactorization algebra of observables defined modulo $\hbar^{k+1}$.
''Lemma.''
//For all open subsets $U \subset M$, the natural map of graded vector spaces//
$$
\ObsStrict^{(k+1)}(U) \to \ObsStrict^{(k)}(U)
$$
//is surjective.//
''Proof of lemma.''
Let us give the set $(i,k) \in \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0}$ the lexicographical ordering, so that $(i,k) > (r,s)$ if $i > r$ or if $i = r$ and $k > s$.
We will let $\ObsStrict^{\le (i,k)}(U)$ be the quotient of $\ObsStrict^{(i)}$ consisting of functionals
$$
O[\Phi] = \sum \hbar^r O_{(r,s)}[\Phi]
$$
satisfying the renormalization group equation and locality axiom as before; but where $O_{(r,s)}[\Phi]$ is only defined for $(r,s) \le (i,k)$.
Similarly, we will let $\ObsStrict^{< (i,k)}(U)$ be the quotient where the $O_{(r,s)}[\Phi]$ are only defined for $(r,s) < (i,k)$.
We will show that the map
$$
\ObsStrict^{\le (i,k)}(U) \rightarrow \ObsStrict^{< (i,k)}(U)
$$
is surjective. The result will follow.
For every parametrix $\Phi$, we will define a map
$$
L_{\Phi} : \ObsStrict^{< (i,k)}(U) \rightarrow \ObsStrict^{\le (i,k)}(M)
$$
with the property that the composed map
$$
\ObsStrict^{<(i,k)}(U) \xto{L_{\Phi}} \ObsStrict^{\le(i,k)}(M) \to \ObsStrict^{<(i,k)}(M)
$$
is the natural inclusion map.
If
$$
O \in \ObsStrict^{< (i,k)}(U)
$$
we will define
$$
L_{\Phi} (O) \in \ObsStrict^{\le (i,k)}(M)
$$
by
$$
L_{\Phi}(O)_{r,s}[\Phi] = \begin{cases} O_{r,s}[\Phi] & \text{ if } (r,s) < (i,k) \\
0 & \text{ if } (r,s) = (i,k)
\end{cases}.
$$
If $\Psi \neq \Phi$, then $L_{\Phi} (O)_{r,s}[\Psi]$ is determined from $L_{\Phi}(O)_{r,s}[\Phi]$ by the renormalization group flow. If $(r,s) < (i,k)$, then
$$
L_{\Phi} (O)_{r,s}[\Psi] = O_{r,s}[\Psi].
$$
But
$$
I_{i,k}[\Psi] + \delta \left( L_{\Phi} (O)_{i,k}\right) [\Psi] = W_{i,k} \left(P(\Psi) - P(\Phi) , I[\Phi] + \delta O_{< (i,k)} [\Phi] \right)
$$
for $\delta$ a parameter of square zero and cohomological degree opposite to that of $O$. Hence $L_{\Phi}(O)_{r,s}$ satisfies the relevant RGE.
To complete the proof of the lemma, we need to show the following sublemma.
''Sublemma.''
//For all $O\in \ObsStrict^{<(i,k)}(U)$, we can find $\Phi$ so that $L_{\Phi} O$ lies in $\ObsStrict^{\le(i,k)}(U) \subset \ObsStrict^{\le(i,k)}(M)$.//
''Proof.''
We need to find $\Phi$ such that for all sufficiently small $\Psi$, the support of $L_{\Phi}(O)$ is in a compact subset of $U^k$. This argument resembles previous support arguments (e.g., the [[product lemma| proof of product lemma]]). The proof of this is a straightforward analysis of the Feynman diagrams appearing in the expression
$$
L_{\Phi}(O)_{i,k}[\Psi] = \sum_{\gamma} \frac{1}{\abs{\op{Aut}(\gamma) } } W_\gamma .
$$
The sum is over all connected Feynman diagrams of genus $i$ with $k$ tails. The edges are labelled by $\Psi - \Phi$. Each graph has one special vertex, where $O[Phi]$ appears. More explicitly, if this vertex is of genus $r$ and valency $s$ it is labelled by $O_{r,s}[\Phi]$. Every other vertex, of genus $a$ and valency $b$, is labelled by $I_{a,b}[\Phi]$. Note that only a finite number of graphs appear in this sum.
By choosing $\Phi$ to be sufficiently small, we can assume that each $O_{r,s}[\Phi]$ has support on a compact subset of $U^s$, and that each $I_{a,b}[\Phi]$ is supported as close as we like to the small diagonal in $M^b$. If we also take $\Psi$ sufficiently small, the propagator $\Psi - \Phi$ has support as close as we like to the small diagonal in $M^2$.
It follows that we can find some small $\Phi$ such that, for all sufficiently small $\Psi$, the weight of $W_\gamma$ of each graph is supported on a compact subset of $U^k$. Since there are only a finite number of graphs in the sum, the result follows.
$\square$
We have [[seen | the prefactorization algebra of strict observables]] how to define a prefactorization algebra of local observables for our quantum field theory. In this page we will show that this prefactorization algebra is in fact a factorization algebra. Further, we will see that this prefactorization algebra is quasi-isomorphic to the factorization algebra $\ObsHomotopy$ of observables satifying the RG equation up to homotopy.
''Theorem.''
* //The prefactorization algebra $\ObsStrict$ of quantum observables is, in fact, a factorization algebra.//
* // There is a quasi-isomorphism of factorization algebras//
$$
\ObsStrict \simeq \ObsHomotopy
$$
* //Further, there is an isomorphism//
$$
\ObsStrict \otimes_{\mathbb{R}[[\hbar]]} \mathbb{R} \cong \Obs^{cl}
$$
//betwen the $\operatorname {mod} \hbar$ reduction of the factorization algebra of quantum observables, and the factorization algebra of classical observables.//
!!! The map from strict observables to homotopy observables
First, let us describe a natural map of prefactorization algebras
$$
\ObsStrict \to \ObsHomotopy.
$$
This map is $\R[[\hbar]]$ linear. Recall that the definition of $\ObsHomotopy$ requires the choice of a family $\Phi_t$ of parametrices depending on $t \in (0,1)$ such that $\Phi_t$ is supported within $t$ of the diagonal. An element of $\ObsHomotopy(U)$ is a functional
$$
O \in \Oo(\E(U)) \otimes \A
$$
where
$$
\A = \colim_{\delta \to 0} \Omega^\ast ( (0,\delta ))
$$
is the space of germs of forms at $0$ in $(0,\delta)$. The differential on $\ObsHomotopy(U)$ implements the renormalization group flow up to homotopy.
If $O \in \ObsStrict(U)$ is an observable satisfying the RG flow up to homotopy, we can consider the family $O[\Phi_t]$ of functionals for $t \in (0,1)$. Because these functionals, for different values of $t$, are related by the renormalization group flow, the family $O[\Phi_t]$ depends smooothly on $t$, and thus can be viewed as an element of
$$
O[\Phi_t] \in \Oo(\E(M)) \otimes \cinfty( (0,1)).
$$
Using the natural map $\cinfty( (0,1)) \to \A$ we turn $O [\Phi_t]$ into an element of $\ObsHomotopy(U)$. as desired. The map
$$
\ObsStrict(U) \to \ObsHomotopy(U)
$$
is easily seen to be a cochain map.
!!! Proof of the theorem
This theorem will be a corollary of a more technical proposition.
''Proposition''
//For any open subset $U \subset M$, let us filter $\ObsStrict(U)$ by saying that $F^k \ObsStrict(U)$ is the sub $\mathbb{R}[[\hbar]]$-module consisting of those observables which are zero modulo $\hbar^k$. Note that this is a filtration by sub prefactorization algebras over the ring $\mathbb{R}[[\hbar]]$.//
//Then, there is an isomorphism of prefactorization algebras//
$$
\operatorname{Gr} \ObsStrict \simeq \Obs^{cl} \otimes_{\mathbb{R}} \mathbb{R}[[\hbar]].
$$
''Proof of the theorem, assuming the proposition.''
The fact that $\ObsStrict$ is a factorization algebra will follow from the fact that the natural map
$$
\ObsStrict \to \ObsHomotopy
$$
is a quasi-isomorphism. Both sides are filtered by powers of $\hbar$, as above. To show that this map is a quasi-isomorphism, it suffices to show that the map
$$
\Gr \ObsStrict \to \Gr \ObsHomotopy
$$
is a quasi-isomorphism. Now, we already know that
$$
\Gr \ObsHomotopy \simeq \Obs^{cl} \otimes \R [[\hbar]],
$$
so the result follows.
$\square$
''Proof of the proposition.''
The first step in the proof of the proposition is the following lemma.
''Lemma.''
//Let $\ObsStrict^{(0)}$ denote the prefactorization algebra of observables which are only defined modulo $\hbar$. Then, there is an isomorphism of prefactorization algebras//
$$
\ObsStrict^{(0)} \simeq \Obs^{cl}
$$
//of differential graded prefactorization algebras.//
''Proof of lemma.''
Suppose that $O \in \Obs^{cl}(U)$ is a classical observable. Thus, $O$ is an element of the cochain complex $\mathcal{O}(\mathcal{E}\mid U)$ of functionals on the space of fields on $U$. From this, we need to produce an element of $\ObsStrict^{(0)} $. An element of $\ObsStrict^{(0)}$ is a collection of functionals $O[\Phi] \in \mathcal{O}(\mathcal{E})$, one for every parametrix $\Phi$, satisfying a classical version of the renormalization group equation, and an axiom saying that $O[\Phi]$ is supported on $U$ for sufficiently small $\Phi$.
Given an element
$$
O \in \Obs^{cl}(U) = \mathcal{O}( \mathcal{E} \mid_U) ,
$$
we define an element
$$
\{ O[\Phi] \} \in \ObsStrict^{(0)}
$$
by the formula
$$
\delta O[\Phi] = \operatorname{lim}_{P' \to 0} W ( P (\Phi) - P(\Psi), I[\Psi] + \delta O[\Psi] ) - I[\Phi] \text{ modulo } \hbar.
$$
The Feynman diagram expansion of the right hand side only involves trees, since we are working modulo $\hbar$. The fact that we are only using trees implies the limit exists.
The map
$$
\Obs^{cl}(U) \to \ObsStrict^{(0)}(U)
$$
is easily seen to be a map of cochain complexes, compatible with the structure of prefactorization algebra present on both sides.
A simple inductive argument shows that this map is an isomorphism.
$\square$
The next (and most difficult) step in the proof of the proposition is the following lemma.
Let $\ObsStrict^{(k)}$ denote the prefactorization algebra of observables defined modulo $\hbar^{k+1}$.
''Lemma.''
//For all open subsets $U \subset M$, the natural map of graded vector spaces//
$$
\ObsStrict^{(k+1)}(U) \to \ObsStrict^{(k)}(U)
$$
//is surjective.//
''Proof of lemma.''
Let us give the set $(i,k) \in \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0}$ the lexicographical ordering, so that $(i,k) > (r,s)$ if $i > r$ or if $i = r$ and $k > s$.
We will let $\ObsStrict^{\le (i,k)}(U)$ be the quotient of $\ObsStrict^{(i)}$ consisting of functionals
$$
O[\Phi] = \sum \hbar^r O_{(r,s)}[\Phi]
$$
satisfying the renormalization group equation and locality axiom as before; but where $O_{(r,s)}[\Phi]$ is only defined for $(r,s) \le (i,k)$.
Similarly, we will let $\ObsStrict^{< (i,k)}(U)$ be the quotient where the $O_{(r,s)}[\Phi]$ are only defined for $(r,s) < (i,k)$.
We will show that the map
$$
\ObsStrict^{\le (i,k)}(U) \rightarrow \ObsStrict^{< (i,k)}(U)
$$
is surjective. The result will follow.
For every parametrix $\Phi$, we will define a map
$$
L_{\Phi} : \ObsStrict^{< (i,k)}(U) \rightarrow \ObsStrict^{\le (i,k)}(M)
$$
with the property that the composed map
$$
\ObsStrict^{<(i,k)}(U) \xto{L_{\Phi}} \ObsStrict^{\le(i,k)}(M) \to \ObsStrict^{<(i,k)}(M)
$$
is the natural inclusion map.
If
$$
O \in \ObsStrict^{< (i,k)}(U)
$$
we will define
$$
L_{\Phi} (O) \in \ObsStrict^{\le (i,k)}(M)
$$
by
$$
L_{\Phi}(O)_{r,s}[\Phi] = \begin{cases} O_{r,s}[\Phi] & \text{ if } (r,s) < (i,k) \\
0 & \text{ if } (r,s) = (i,k)
\end{cases}.
$$
If $\Psi \neq \Phi$, then $L_{\Phi} (O)_{r,s}[\Psi]$ is determined from $L_{\Phi}(O)_{r,s}[\Phi]$ by the renormalization group flow. If $(r,s) < (i,k)$, then
$$
L_{\Phi} (O)_{r,s}[\Psi] = O_{r,s}[\Psi].
$$
But,
$$
I_{i,k}[\Psi] + \delta \left( L_{\Phi} (O)_{i,k}\right) [\Psi] = W_{i,k} \left(P(\Psi) - P(\Phi) , I[\Phi] + \delta O_{< (i,k)} [\Phi] \right)
$$
(if $\delta$ is a parameter of square zero and cohomological degree opposite to that of $O$).
To complete the proof of the lemma, we need to show the following.
''Lemma.''
//For all $O\in \ObsStrict^{<(i,k)}(U)$, we can find $\Phi$ so that $L_{\Phi} O$ lies in $\ObsStrict^{\le(i,k)}(U) \subset \ObsStrict^{\le(i,k)}(M)$.//
''Proof.''
We need to find $\Phi$ such that for all sufficiently small $\Psi$, the support of $L_{\Phi}(O)$ is in a compact subset of $U^k$. The proof of this is a straightforward analysis of the Feynman diagrams appearing in the expression
$$
L_{\Phi}(O)_{i,k}[\Psi] = \sum_{\gamma} \frac{1}{\abs{\op{Aut}(\gamma) } } W_\gamma .
$$
The sum is over all connected Feynman diagrams of genus $i$ with $k$ tails. The edges are labelled by $\Psi - \Phi$. Each graph has one special vertex; if this vertex is of genus $r$ and valency $s$ it is labelled by $O_{r,s}[\Phi]$. Every other vertex, of genus $a$ and valency $b$, is labelled by $I_{a,b}[\Phi]$. Note that only a finite number of graphs appear in this sum.
By choosing $\Phi$ to be sufficiently small, we can assume that each $O_{r,s}[\Phi]$ has support on a compact subset of $U^s$, and that each $I_{a,b}[\Phi]$ is supported as close as we like to the small diagonal in $M^b$. If we also take $\Psi$ sufficiently small, the propagator $\Psi - \Phi$ has support as close as we like to the small diagonal in $M^2$.
It follows that we can find some small $\Phi$ such that, for all sufficiently small $\Psi$, the weight of $W_\gamma$ of each graph is supported on a compact subset of $U^k$. Since there are only a finite number of graphs in the sum, the result follows.
$\square$
Recall the following definition.
''Definition''
//A $P_0$ algebra (in the category of cochain complexes) is a commutative differential graded algebra together with a Poisson bracket $\{-,-\}$ of cohomological degree $1$, which satisfies the Jacobi identity and the Liebniz rule.//
We will show that the commutative factorization algebra $\Obs^{cl}$ of observables of a classical field theory has the structure of a factorization algebra valued in lax $P_0$ algebras. This means that, to each open subset $U \subset M$, we construct a lax $P_0$ algebra in the symmetric monoidal category of nuclear spaces. Recall that a [[lax algebra | lax algebras over an operad]] over an operad consists of a sequence of cochain complexes $V(n)$, for each $n$, with various structures including quasi-isomorphisms
$$
V(n) \to V(1)^{\otimes n}.
$$
Lax algebras over a Hopf operad $P$ form a symmetric monoidal category. In particular, lax $P_0$ algebras form a symmetric monoidal category. The tensor product is defined by saying that
$$
(V \otimes W)(n) = V(n) \otimes W(n).
$$
''Definition''. //A lax $P_0$ factorization algebra is a prefactorization algebra $\F$ with values in the symmetric monoidal category of lax $P_0$ algebras (in nuclear spaces); with the property that the prefactorization algebra//
$$U \mapsto \F(1)(U)$$
//is a factorization algebra.//
We can state the theorem as follows.
''Theorem''. //For any [[classical field theory | definition of classical field theory]] on $M$, there is a lax $P_0$ factorization algebra $\til{\Obs}^{cl}$, together with a quasi-isomorphism of factorization algebras//
$$
\til{\Obs}^{cl}(1) \iso \Obs^{cl}.
$$
The idea of the definition is very simple. Let us start with a finite dimensional model. Let $\g$ be an $L_\infty$ algebra equipped with an invariant antisymmetric element $P \in \g \otimes \g$, of cohomological degree $3$. This can be viewed (according to the correspondence between [[formal moduli problems and Lie algebras]]) as a bivector on $B \g$, and so defines a Poisson bracket on $\Oo(B \g) = C^\ast(\g)$. Concretely, this Poisson bracket is defined, on the generators $\g^\vee[-1]$ of $C^\ast(\g)$, to be the tensor $P$ viewed as a map
$$
\g^\vee \otimes \g^\vee \to \R.
$$
Now, let $L$ is an elliptic $L_\infty$ algebra describing a classical field theory. Then, the kernel for the isomorphism $\L(U) \iso \L^!(U)[-3]$ is an element $P \in \br{\L}(U) \otimes \br{\L}(U)$, which is symmetric, invariant, and of degree $3$.
We would like to use this idea to define the Poisson bracket on
$$
\Obs^{cl}(U) = C^\ast (\L(U)).
$$
As in the finite dimensional case, in order to define such a Poisson bracket we would need an invariant tensor in $\L(U)^{\otimes 2}$. Our tensor is instead in $\br{\L}(U)^{\otimes 2}$, which contains $\L(U)^{\otimes 2}$ as a dense subspace.
We solve this problem by finding a subcomplex
$$
\til{\Obs}^{cl}(U) \subset \Obs^{cl}(U)
$$
on which the Poisson bracket is well-defined, and such that the inclusion is a homotopy equivalence. We run into an additional technical problem: the Poisson bracket is defined using the [[inductive tensor product | completed inductive tensor product]] $\br{\otimes}$, and not the projective tensor product $\otimes$. This is why we find a lax $P_0$ algebra.
!!! Overview
Here is an overview of this section.
* [[The Poisson structure for free field theories]]
In this page, we give a simple direct construction of the $P_0$ structure on the observables for a free field theory. For this class of field theories, we can find a model for the classical observables which has a strict (as opposed to lax) $P_0$ structure.
* [[The Poisson structure for cotangent field theories]]
This page gives a construction of the $P_0$ structure in another special case, that of cotangent field theories. (Reading this page is not necessary for understanding the general construction).
* [[The Poisson structure for a general classical field theory]]
In this page we construct the Poisson structure in general.
The model problems of classical and quantum mechanics involve a particle moving in some Euclidean space $\mathbb{R}^n$ under the influence of some fixed field. Our main goal in this page is to describe these model problems in a way that makes the idea of a [[factorization algebra]] emerge naturally, but we also hope to give mathematicians some feeling for the physical meaning of terms like "field" and "observable." We will not worry about making precise definitions, since that's what this site aims to do. As a narrative strategy, we describe a kind of cartoon of a physical experiment, and ask that physicists accept this cartoon as a friendly caricature elucidating the features of physics we most want to emphasize.
!! A particle in a box
For the general framework we want to present, the details of the physical system under study are not so important. However, for concreteness, we will focus attention on a very simple system: that of a single particle confined to some region of space. As our particle We confine our particle inside some box and occasionally take measurements of this system. The set of possible trajectories of the particle around the box constitute all the imaginable behaviors of this particle; we might write this mathematically as $Maps(I,box)$, where $I \subset \mathbb{R}$ denotes the time interval over which we conduct the experiment. In other words, the set of possible behaviors forms a space of //fields// on the timeline of the particle.
The behaviour of our theory is governed by the action functional. The simplest case is the action of the massless free field theory, whose value on a function $f: I \to box$ is
$$
S(f) = \int_{I} \ip{ f , \Delta f }.
$$
The aim of this page is to outline the structure one would expect the observables -- that is, the possible measurements one can make -- should satisfy.
!!! Classical mechanics
Let us start by considering the much simpler case, where our particle is treated as a classical system. In that case, the trajectory of the particle is constrained to be in a solution to the ~Euler-Lagrange equations of our theory. For example, if the action functional governing our theory is that of the massless free theory, then a map $f : I \to box$ satisfies the ~Euler-Lagrange equation if it is a straight line.
We are interested in the observables for this classical field theory. Since the trajectory of our particle is constrained to be a solution to the ~Euler-Lagrange equation, the only measurements one can make are functions on the space of solutions to the ~Euler-Lagrange equation.
If $U \subset \mathbb{R}$ is an open subset, we will let $ \operatorname{Fields (U)} $ denote the space of fields on $ U $, that is, the space of maps $ f : U \rightarrow box $. We will let
$$
\operatorname{EL}(U) \subset \operatorname{Fields }(U)
$$ denote the subspace consisting of those maps $f : U \to box$ which are solutions to the ~Euler-Lagrange equation. As $U$ varies, $\operatorname{EL}(U)$ forms a sheaf of spaces on $\mathbb{R}$.
We will let $\operatorname{Obs}^{cl}(U)$ denote the space of functions on $\operatorname{EL}(U)$ (the precise class of functions we will consider will be discussed later). As $U$ varies, the spaces $\operatorname{Obs}^{cl}(U)$ form a cosheaf of commutative algebras on $\mathbb{R}$. We will think of $ \operatorname{Obs}^{cl}(U) $ as the space observables for our classical system which only consider the behaviour of the particle on times contained in $ U $.
Note that $\Obs^{cl}(U)$ is a cosheaf of commutative algebras on $\R$.
!!! Measurements in quantum mechanics
The notion of measurement is fraught in quantum theory, but we will take a very concrete view. Taking a measurement means that we have physical measurement device (e.g., a camera) that we allow to interact with our system for a period of time. The measurement is then how our measurement device has changed due to the interaction. In other words, we //couple// the two physical systems, then decouple them and record how the measurement device has modified from its initial condition. (Of course, there is a symmetry in this situation: both systems are affected by their interaction, so a measurement inherently disturbs the system under study.)
The //observables// for a physical system are all the imaginable measurements we could take of the system. Instead of considering all possible observables, we might also consider those observables which occur within a specified time period. This period can be specified by an open interval $U \subset \R$.
Thus, we arrive at the following principle.
<<<
''Principle 1''. For every open subset $U \subset \R$, we have a set $\Obs(U)$ of observables one can make on $U$.
<<<
The superposition principle tells us that quantum mechanics (and quantum field theory) is fundamentally linear. This leads to
<<<
''Principle 2''. The set $\Obs(U)$ is a complex vector space.
<<<
We think of $\Obs(U)$ as being the space of ways of coupling a measurement device to our system on the region $U$. Thus, there is a natural map $\Obs(U) \to \Obs(V)$ if $U \subset V$ is an open subset. This means that the space $\Obs(U)$ forms a pre-cosheaf.
!!! Combining observables
Measurements (and so observables) differ qualitatively in the classical and quantum settings. If we study a classical particle, the system is not noticeably disturbed by measurements, and so we can do multiple measurements at the same time. Hence, on each interval $J$ we have a commutative multiplication map $Obs(J) \otimes Obs(J) \rightarrow Obs(J)$, as well as the maps that let us combine observables on disjoint intervals.
For a quantum particle, however, a measurement disturbs the system significantly. Taking two measurements simultaneously is incoherent, as the measurement devices are coupled to each other and thus also affect each other, so that we are no longer measuring just the particle. Quantum observables thus do not form a cosheaf of commutative algebras on the interval. However, there are no such problems with combining measurements occuring at different times. Thus, we find the following.
<<<
''Principle 3''. If $U, U'$ are disjoint open subsets of $\R$, and $U, U' \subset V$ where $V$ is also open, then there is a map
$$
\star: \Obs(U) \otimes \Obs(U') \to \Obs(V).
$$
If $O \in \Obs(U)$ and $O' \in \Obs(U')$, then $O \star O'$ is defined by coupling our system to measuring device $O$ for $t \in U$, and to device $O'$ for $t \in U'$.
Further, these maps are commutative, associative, and compatible with the maps $\Obs(U) \to \Obs(V)$ associated to inclusions $U \subset V$ of open subsets. (The precise meaning of these terms is detailed [[here | prefactorization algebras]]).
<<<
!!! Perturbative theory and the correspondence principle
In the bulk of this paper, we will be considering perturbative quantum theory. For us, this means that we work over the base ring $\C[[\hbar]]$, where at $\hbar = 0$ we find the classical theory. In perturbative theory, therefore, the space $\Obs(U)$ of observables on an open subset $U$ is a $\C[[\hbar]]$ module, and the product maps are $\C[[\hbar]]$-linear.
The correspondence principle states that the quantum theory, in the $\hbar \to 0$ limit, must reproduce the quantum theory. Applied to observables, this leads to the following principle.
<<<
''Principle 4''. The vector space $\Obs^q(U)$ of quantum observables is a flat $\C[[\hbar]]$ module which, modulo $\hbar$, is the space $\Obs^{cl}(U)$ of classical observables.
<<<
These simple principles are at the heart of our approach to quantum field theory. They say, roughly, that the observables of a quantum field theory form a factorization algebra, which is a quantization of the factorization algebra associated to a classical field theory. The main theorem presented in this paper is that one can use the techniques of perturbative renormalization to construct factorization algebras perturbatively quantizing a certain class of classical field theories (including many classical field theories of physical and mathematical interest).
!!! Associative algebras in quantum mechanics
The principles we have described so far indicate that the observables of a quantum mechanical system should assign, to every open subset $U \subset \R$, a vector space $\Obs(U)$, together with a product map
$$
\Obs(U) \otimes \Obs(U') \to \Obs(V)
$$
if $U,U'$ are disjoint open subsets of an open subset $V$. This is the basic data of a [[factorization algebra | prefactorization algebras]].
It turns out that the factorization algebra produced by our quantization procedure applied to quantum mechanics has a [[special property | locally constant factorization algebras]], which ensures that the vector spaces $\Obs ((a,b))$, for various intervals $(a,b)$, are canonically isomorphic. Let us denote this vector space by $A$.
The product map
$$
\Obs((a,b)) \otimes \Obs((c,d)) \to \Obs((a,d))
$$
(defined when $a < b < c < d$) becomes, when we perform this identification, a product map
$$
m : A \otimes A \to A.
$$
The axioms of a factorization algebra imply that this multiplication maps $A$ into an associative algebra.
This should be familiar to topologists: associative algebras are algebras over the operad of little intervals in $\R$, and this is precisely what we have described. (As we will see [[later| the weyl algebra and the free field on the real line ]], this associative algebra is the Weyl algebra one expects to find as the algebra of observables of quantum mechanics).
One important point to take away from this discussion is that //associative algebras appear in quantum mechanics because associative algebras are connected with the geometry of $\R$//. There is no fundamental connection between associative algebras and any concept of "quantization": associative algebras only appear when one considers one-dimensional quantum field theories. As we will see later, when one considers quantum field theories on $n$-dimensional space times, one finds a structure reminiscent of an $E_n$-algebra instead of an $E_1$-algebra.
We will show that the category of factorization algebras has a natural enhancement to a multicategory, enriched in simplicial sets. This will allow us to define what it means for a topological operad to act on a factorization algebra. This definition will then be extended to define what it means for any differential graded Hopf operad (i.e., operad in the category of dg cocommutative coalgebras) to act on a factorization algebra.
!!! Prefactorization algebras as a symmetric monoidal category
The category of prefactorization algebras is a symmetric monoidal category: if $\mc{F}, \mc{F}'$ are prefactorization algebras, their tensor product $\mc{F} \otimes \mc{F}'$ is defined by
$$
(\mc{F} \otimes \mc{F}') (U) = \mc{F}(U) \otimes \mc{F}'(U).
$$
If $R$ is a commutative dga, and $\mc{F}$ is a prefactorization algebra, we will define a prefactorization algebra $\F \otimes R$ by
$$
\F \otimes R (U) = \F(U) \otimes R.
$$
If $\F, \F'$ are prefactorization algebras, we can define a functor
$$
\begin{split}
\Hom_{\op{PreFact}} (\F, \F' ) : \op{cdgas} & \to \op{Sets}\\
\Hom_{\op{PreFact}} (\F, \F' ) (R) & = \Hom_{\op{PreFact}} (\F, \F' \otimes R ).
\end{split}
$$
By specializing this functor to the commutative algebras $R = \Omega^\ast(\Delta^n)$ of forms on the $n$-simplex, we see that $\Hom_{\op{PreFact}} (\F, \F')$ is, in particular, a simplicial set.
There is no reason, however, that this tensor product should preserve the subcategory of factorization algebras. Thus, we do not know how to make the category of factorization algebras into a symmetric monoidal category. However, we will see that factorization algebras form a multicategory.
!!! The multicategory of factorization algebras
Let $\mc{F}_1,\ldots, \mc{F}_n, \mc{F}_{n+1}$ be factorization algebras on a manifold $M$. Let us define
$$
\Hom_{\op{Fact}_M} (\mc{F}_1,\ldots, \mc{F}_n; \mc{F}_{n+1} ) = \Hom_{\op{PreFact}_M} ( \mc{F}_1 \otimes \cdots \otimes \mc{F}_n, \mc{F}_{n+1} ) .
$$
With this definition of morphism, factorization algebras on $M$ form a multicategory. This multicategory $\op{Fact}_M$ is a full sub-multicategory of the multicategory associated to the symmetric monoidal category of prefactorization algebras on $M$.
In this page we will see that the [[local quantum observables]] of a [[quantum field theory]] on a manifold $ M $ form a prefactorization algebra.
The definition of local observables makes it clear that they form a pre-cosheaf: there are natural injective maps of cochain complexes
$$
\Obs^\ast(U) \rightarrow \Obs^\ast(U')
$$
if $ U \subset U' $ is an open subset.
Let $ U $, $ V $ be disjoint open subsets of $ M $. The structure of prefactorization algebra on the local observables is specified by the pre-cosheaf structure mentioned above, and a product map
$$
\Obs^\ast(U) \otimes \Obs^\ast(V) \rightarrow \Obs^\ast(U \amalg V).
$$
This product map needs to be a map of cochain complexes which is associative, commutative, and compatible with the pre-cosheaf structure.
!!!! Defining the product map
Suppose that $O \in \Obs^k(U)$ and $O' \in \Obs^l(V)$ are observables on $ U $ and $ V $ respectively. Note that $O[\Phi]$ and $O'[\Phi]$ are elements of the space $\mathcal{O}(\mathcal{E})$, which is a commutative algebra. In the definition of the prefactorization product, we will use the product of $O[\Phi]$ and $O'[\Phi]$ taken in the commutative algebra $\mathcal{O}(\mathcal{E})$. This product will be denoted $O[\Phi] \ast O'[\Phi] \in \mathcal{O}(\mathcal{E})$.
''Lemma.''
//Let $\delta$ be a square-zero parameter of degree $-k-l$. For all parametrices $\Psi$, and all $i$ and $k$, there is a well-defined limit//
$$
\lim_{\Phi \to 0} W_{i,k}(P(\Psi) - P(\Phi), I[\Phi] + \delta O[\Phi] \ast O'[\Phi]),
$$
//since the term eventually becomes constant for $\Phi$ sufficiently small. (Here, as in \cite{webbook}, the subscript $i,k$ denotes the coefficient of $\hbar^i$ in the expression which is of order $k$ as a function on the space of fields).//
//This implies that there is a well-defined element $\til{O} \in \Obs^{k+l}(U \amalg V)$ such that//
$$
\lim_{\Phi \to 0} W_{i,k}(P(\Psi) - P(\Phi), I[\Phi] + \delta O[\Phi] \ast O'[\Phi]) = I[\Psi] + \delta \til{O} [\Psi].
$$
//We define the product $OO'$ to be $\til{O}$//.
//Further, the resulting map//
$$
\ObsStrict^{k}(U) \otimes \Obs^{l}(V) \to \Obs^{k+l}(U \amalg V)
$$
$$
O \otimes O' \mapsto OO'
$$
//is a cochain map, which gives $\Obs^{\ast}(U)$ the structure of a differential graded prefactorization algebra valued in the symmetric monoidal category of topological $\mathbb{R}[[\hbar]]$ modules, flat over $\mathbb{R}[[\hbar]]$. We call this prefactorization algebra $\Obs$.//
[[Proof of product lemma]]
Now we have the definition of the algebraic structure associated to classical field theory, one can ask whether there is a quantization. A definition of quantum field theory on a manifold $ M$ is given in \cite{webbook}. There, an obstruction-theoretic framework is described for analyzing quantizations of a classical field theory.
We will show that any quantum field theory on $ M $ in the sense of \cite{webbook} can be turned into a factorization algebra on $ M $.
''Theorem''
//Let us fix a classical field theory on $M$ (given by some space of fields, which is assumed to be sections of a graded vector bundle; and a local action functional, whose quadratic part must satisfy certain ellipticity conditions).//
# //The observables of this classical field theory define a $P_0$ factorization algebra $\Obs^{cl}$ on $M$.//
# //Associated to a quantization of this classical theory, in the sense of \cite{webbook}, there is a factorization algebra $\Obs^q$ of quantum observables, over the ring $\R[[\hbar]]$.//
# //This factorization algebra is a quantization of $\Obs^{cl}$. This means that the bracket on $H^\ast (\Obs^{cl}(U))$ arising from $\Obs^{q}$ coincides with the one give by the $P_0$ structure on $\Obs^{cl}(U)$.//
All factorization algebras appearing in this theorem take values in the symmetric monoidal category of cochain complexes of nuclear spaces.
//Remark//: Later we will consider a refined notion | of quantization, where $\Obs^q$ is equipped with a [[certain operadic structure | Beilinson-Drinfeld operad]]. We conjecture that do not show that all quantizations, in the refined sense, arise in this way.
!!! Obstruction theory
It is worth describing the obstruction theory developed in \cite{webbook}. The space $\mathcal{O}_l(\mathcal {E})$ of local action functionals on $\mathcal {E}$ -- that is, of possible Lagrangians on the space of fields $\mathcal {E}$ -- is a cochain complex in a natural way. In the ~Batalin-Vilkovisky picture, the differential is given by bracketing with the classical action $S$.
The zeroth cohomology, $H^0 (\mathcal{O}_l(\mathcal {E}))$, describes equivalence classes of first order deformations of the classical action $S$, satisfying the relevant gauge symmetry conditions. The first cohomology group $H^1 (\mathcal{O}_l(\mathcal {E}))$ describes the obstructions to deforming the classical action in a way preserving the gauge symmetries. The first negative cohomology group $H^{-1} (\mathcal{O}_l(\mathcal {E}))$ describes the infinitesimal automorphisms of the classical field theory described by $(\mathcal {E},S)$. The higher negative cohomology groups correspond to automorphisms of automorphisms, etc.
In \cite{webbook} the following result is proved.
''Theorem.''
//Suppose we have a quantization of the classical field theory defined by $ (\mathcal{E},S) $ to a quantum theory defined modulo $\hbar^{n+1}$. Then the obstruction to lifting this to a quantization defined modulo $\hbar^{n+2}$ is an element of $H^1 (\mathcal{O}_l(\mathcal {E}))$. If this obstruction vanishes, then the space of possible lifts is a torsor for $H^0 (\mathcal{O}_l(\mathcal {E}))$ (more precisely, the simplicial set of possible lifts is a torsor for the simplicial abelian group $H^{\le 0} (\mathcal{O}_l(\mathcal {E}))$).//
This theorem is a mathematical encoding of the renormalization procedure, and is proved using the physicists toolbox of counterterms, Feynman graphs, etc. Together with the previous theorem, this result allows one to produce, starting from a classical Lagrangian, an algebraic object -- a factorization algebra -- which encodes the correlation functions of the quantum field theory. As is usual, there is an ambiguity in going from the classical Lagrangian to the quantum theory; at each order in $\hbar$, we are free to add on an arbitrary term to the classical Lagrangian. However, for //renormalizable// classical Lagrangians, an extra symmetry -- that of rescaling of space-time -- restricts greatly the possible quantizations of a given classical theory (strictly speaking, this symmetry only holds up to terms logarithmic in the scale). This point of view on renormalizability is detailed in \cite{webbook}. In good cases, the space of possible quantizations is parametrized by a finite number of coupling constants.
Pure ~Yang-Mills theory fits into this class of theories. In \cite{webbook}, the deformation-obstruction complex for quantizing pure ~Yang-Mills theory on $\mathbb{R}^4$ was calculated. It was shown that if we restrict attention to quantizations satisfying an additional scaling symmetry, present by virtue of renormalizability of the classical Lagrangian, then the obstruction group vanishes. Further, if we take coefficients in a simple Lie algebra, the deformation group is $1$ dimensionsal. This means that the moduli space of possible quantizations is isomorphic to $\hbar \mathbb{R} [[\hbar]]$; so that the space of theories is parametrized by a single $\hbar$ dependent coupling constant.
Thus, we find
''Corollary.''
//There is a factorization algebra on any $4$-manifold $M$ with a flat metric, which encodes pure ~Yang-Mills theory on $M$.//
We have [[seen|the weak quantization theorem]] how the observables of a quantum field theory are a quantization, in a weak sense, of the lax $P_0$ algebra of observables of a quantum field theory. The definition of quantization appearing in this theorem is unsatisfactory, however, because the bracket on the classical observables arising from the quantum observables is not a Poisson observable.
In this section we will explain a stronger quantization theorem, which, however, relies on more operadic machinery. We should caution the reader that the strong quantization theorem is still work-in-progress, and we do not present all relevant homotopical definitions. In particular, the crucial notion of homotopy BD algebras is not yet written.
''Definition'' //A //BD algebra// is a cochain complex $A$, flat over $\R[[\hbar]]$, equipped with a commutative product and a Poisson bracket of cohomological degree $1$, satisfying the identity//
$$
\d (a\cdot b) = a \cdot (\d b) \pm (\d a ) \cdot b + \hbar \{a,b\}.
$$
The BD operad is investigated in detail [[here | Beilinson-Drinfeld operad]]. Note that, modulo $\hbar$, a BD algebra is a $P_0$ algebra.
''Definition'' //A //quantization// of a $P_0$ algebra $A^{cl}$ is a BD algebra $A^q$, flat over $\R[[\hbar]]$, together with an equivalence of $P_0$ algebras between $A^q / \hbar$ and $A^{cl}$. //
In our strong quantization theorem, we need to use the more flexible notion of [[homotopy BD agebra]]. Our notion of homotopy $P_0$ algebra is sufficiently flexible that it includes lax $P_0$ algebras as a special case.
''Definition'' //A //homotopy quantization// of a homotopy $P_0$ algebra $A^{cl}$ is a homotopy BD algebra $A^q$, flat over $\R[[\hbar]]$, with an equivalence of homotopy $P_0$ algebras $A^q / \hbar \simeq A^{cl}$. //
!!! Strong quantization theorem
Now suppose that we have a classical field theory on $M$, and a BV quantization of the theory. Then, $\Obs^{cl}$ forms a lax $P_0$ factorization algebra on $M$, and so it forms a factorization algebra in homotopy $P_0$ algebras. The strong quantization theorem (which is still work-in-progress) is as follows.
''Theorem'' //Suppose we have a classical field theory on $M$, and a BV quantization of the theory. Then, $\Obs^q$ has a natural structure of homotopy BD factorization algebra (in particular, for every $U \subset M$, the quantum observables $\Obs^{q}(U)$ is a homotopy BD algebra). Further, $\Obs^q(U)$ is a homotopy quantization of the homotopy $P_0$ algebra $\Obs^{cl}(U)$.//
The proof of this result relies on a characterization of a homotopy BD algebra by its "functor of points", which is a functor on Artinian BD algebras. This characterization is work-in-progress of Josh Shadlen.
We have explained how a classical field theory gives rise to a lax $P_0$ factorization algebra $\Obs^{cl}$, and how a quantum field theory (in the sense of \cite{webbook}) gives rise to a factorization algebra $\Obs^{q}$ over $\R[[\hbar]]$, which specializes at $\hbar = 0$ to the factorization algebra $\Obs^{cl}$ of classical observables. In this section we will state our //weak quantization theorem//, which says that the Poisson bracket on $\Obs^{cl}$ is compatible, in a certain sense, with the quantization given by $\Obs^{q}$.
This statement is the analog, in our setting, of a familiar statement in quantum-mechanical deformation quantization. [[Recall | Introduction]] that in that setting, we require that the associative product on the algebra $A^{q}$ of quantum observables is related to the Poisson bracket on the Poisson algebra $A^{cl}$ of classical observables by the formula
$$
\{a,b\} = \lim_{\hbar \to 0} \hbar^{-1} [\til{a}, \til{b} ]
$$
where $\til{a}, \til{b}$ are any lifts of the elements $a,b \in A^{cl}$ to $A^q$.
One can make a similar definition in the world of $P_0$ algebras. If $A^{cl}$ is any commutative differnetial graded algebra, and $A^{q}$ is a cochain complex flat over $\R[[\hbar]]$ which reduces to $A^{cl}$ modulo $\hbar$, then we can define a cochain map
$$
\{-,-\}_{A^q} : A^{cl} \otimes A^{cl} \to A^{cl}
$$
which measures the failure of the commutative product on $A^{cl}$ to lift to a product on $A^{q}$, to first order in $\hbar$. (A precise definition is given [[here|P_0 operad]]).
Now, suppose that $A^{cl}$ is a $P_0$ algebra (that is, a commutative dga equipped with a Poisson bracket of cohomological degree $1$). Let $A^{q}$ be a cochain complex flat over $\R[[\hbar]]$ which reduces to $A^{cl}$ modulo $\hbar$. We say that $A^{q}$ is a //weak quantization// of $A^{cl}$ if the bracket $\{-,-\}_{A^{q}}$ on $A^{cl}$, induced by $A^{q}$, is homotopic to the given Poisson bracket on $A^{cl}$.
This is a very weak notion, because the bracket $\{-,-\}_{A^q}$ on $A^{cl}$ need not be a Poisson bracket; it is simply a bilinear map. When we discuss the notion of [[strong quantization|the strong quantization theorem]], we will explain how to put a certain operadic structure on $A^{q}$ which guarantees that this induced bracket is a Poisson bracket.
!!! The weak quantization theorem
Now that we have the definition of weak quantization at hand, we can state our weak quantization theorem.
For every open subset $U \subset M$, $\Obs^{cl}(U)$ is a lax $P_0$ algebra. Given a BV quantization of our classical field theory, $\Obs^{q}(U)$ is a cochain complex flat over $\R[[\hbar]]$ which coincides, modulo $\hbar$, with $\Obs^{cl}(U)$. Our definition of weak quantization makes sense with minor modifications for lax $P_0$ algebras as well as for ordinary $P_0$ algebras.
''Theorem'' (Weak quantization)
//For every $U \subset M$, the cochain complex $\Obs^{q}(U)$ of classical observables on $U$ is a weak quantization of the lax $P_0$ algebra $\Obs^{cl}(U)$. //
!!! Guide to reading
This theorem is the main goal of the section on quantum field theory. The details are provided in part (h) of that section.
/***
|Name|TiddlerPasswordPlugin|
|Source|http://www.TiddlyTools.com/#TiddlerPasswordPlugin|
|Documentation|http://www.TiddlyTools.com/#TiddlerPasswordPluginInfo|
|Version|1.1.3|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|block viewing of tiddler content by prompting for a password before content is displayed|
This plugin blocks viewing of specific tiddler content by prompting for a NON-SECURE, UNENCRYPTED password before the tiddler is displayed. If the correct password is not entered, the tiddler is automatically closed. The process does not prevent tiddler content from being viewed directly from the TiddlyWiki source file's storeArea, nor does it encrypt the tiddler content in any way. Because it is relatively simple to bypass and/or disable the password prompting process, this macro should be thought of as a "latch" rather than a "lock" on a given tiddler.
!!!!!Documentation
> see [[TiddlerPasswordPluginInfo]]
!!!!!Installation Notes
<<<
''As soon as you have installed this plugin, you should change the default admin password in [[TiddlerPasswordPluginConfig]].'' Note: the configuration tiddler is password-protected to prevent the admin password from being viewed (and/or modified) unless the current password is provided. By default, the admin password is set to "admin".
<<<
!!!!!Revisions
<<<
2008.03.10 [*.*.*] plugin size reduction - documentation moved to [[TiddlerPasswordPluginInfo]]
2007.09.13 [1.1.3] adjusted wording of "cancelMsg" text so it can apply to either view-mode or edit-mode activities, and documented usage in ViewTemplate/EditTemplate.
| Please see [[TiddlerPasswordPluginInfo]] for previous revision details |
2006.12.02 [1.0.0] initial release - converted from GetTiddlerPassword inline script
<<<
!!!!!Code
***/
//{{{
version.extensions.TiddlerPasswordPlugin= {major: 1, minor: 1, revision: 3, date: new Date(2007,9,13)};
config.macros.getTiddlerPassword = {
msg: "Please enter a password to view '%0'",
defaultText: "enter password here",
retryMsg: "'%0' is not the correct password for '%1'. Please try again:",
cancelMsg: "Sorry, you cannot access '%0' without a valid password.",
thanksMsg: "Thank you, your password has been accepted.",
handler: function(place,macroName,params,wikifier,paramString,tiddler) {
var here=story.findContainingTiddler(place); if (!here) return;
var title=tiddler?tiddler.title:here.getAttribute("tiddler");
var who=here.getAttribute("logID");
var userPass=params[0]?params[0]:""; if (userPass=='-') userPass="";
var msg=params[1]?params[1]:this.msg;
if (who==userPass||who==this.adminPass) return; // already 'logged in'?
var who=prompt(msg.format([title]),this.defaultText); // ask for ID
while (who && who!=userPass && who!=this.adminPass) // not correct ID?
who=prompt(this.retryMsg.format([who,title]),this.defaultText); // ask again
if (who==userPass||who==this.adminPass) // correct ID? mark tiddler logged in...
{ here.setAttribute("logID",who); alert(this.thanksMsg); }
else // incorrect ID (e.g., entry cancelled by user)...
{ story.closeTiddler(here.getAttribute("tiddler")); alert(this.cancelMsg.format([title])); }
}
}
// default admin password (may be overridden in TiddlerPasswordPluginConfig)
if (config.macros.getTiddlerPassword.adminPass==undefined)
config.macros.getTiddlerPassword.adminPass="admin";
//}}}
// // Tiddler Admin Password Configuration... <<getTiddlerPassword>> /% rest of tiddler will not be displayed without password... %/
//{{{
config.macros.getTiddlerPassword.adminPass="Q1gdRRzz";
//}}}
// {{small{NOTE: after changing the password, save-and-reload the document for the change to take effect}}} //
Higher and derived stacks: a global overview arXiv:math/0604504
|~ViewToolbar|closeTiddler closeOthers +editTiddler > fields syncing permalink references jump|
|~EditToolbar|+saveTiddler -cancelTiddler deleteTiddler|
''Definition.'' //A topological cochain complex is a sequence $E_i$ of locally convex Hausdorff topological vector spaces, for $i \in \Z$, together with continuous linear maps $\d: E_i \to E_{i+1}$ such that $\d \circ \d = 0$. //
Very often, our topological vector spaces are also [[nuclear|overview of nuclear spaces]]. We will call a cochain complex of nuclear spaces a //nuclear cochain complex//. One can define cochain maps and cochain homotopies between cochain complexes of topological vector spaces in the usual way (using, of course, continuous linear maps).
!!! Filtered topological cochain complexes
''Definition''.// Let $E$ be a topological cochain complex. A //topologically split complete decreasing filtration// on $E$ is a filtration of $E$ by descending closed subcomplexes $F^i E$, for $i \in \Z_{\ge 0}$, such that//
# //$E = \lim_{i \in \Z_{\ge 0}} E / F^i E$ (as topological cochain complexes).//
# //The filtration $F^i E$ is topologically split. Thus, if we let $\Gr^i E = F^i E / F^{i+1} E$, there is an isomorphism of graded topological vector spaces $E = \prod_i \Gr^i E.$ (This isomorphism need not respect the differentials)//.
''Lemma.'' //Let $E$, $G$ be topological cochain complexes equipped with a topologically split complete decreasing filtrations $F^i E$, $F^i G$. Let $f : E \to G$ be a filtration preserving cochain map. Suppose that//
$$
\Gr^i f : \Gr^i E \to \Gr^i G
$$
//is a homotopy equivalence. Then so is $f$.//
[[Proof| proof of homotopy equivalence lemma]]
!!! Special classes
We are occassionally interested in special classes of topological vector spaces. Let $P$ denote any full subcategory of the category topological vector spaces (for instance, $P$ could denote Frechet spaces or nuclear spaces). We say a topological cochain complex is a homotopy $P$-complex if it is homotopy equivalence to a complex of topological vector spaces in $P$.
''Lemma.'' //Let $P$ be any fully subcategory of the category of topological vector spaces which is closed under countable products. Let $E$ be a topological cochain complex equipped with a topologically split complete decreasing filtration $F^i E$. Suppose that $\Gr^i E$ is a homotopy $P$-complex. Then so is $E$. //
[[Proof | proof of cochain homotopy type lemma]]
!!! Tensor products
''Definition''. //Let $E, F$ be topological cochain complexes. Define $E \otimes F$ by//
$$
(E \otimes F)^k = \prod_{i + j = k} E^i \otimes F^j.
$$
//Define $E \br{\otimes} F$ (the [[completed inductive tensor product]] of $E$ and $F$) by//
$$
(E \br{\otimes} F)^k = \prod_{i + j = k} E^i \br{\otimes} F^j.
$$
<<getTiddlerPassword>>
Although the focus on this wiki is quantum field theory, factorization algebras apply beautifully to the setting of topological quantum field theories. Perhaps the most attractive feature is that "extending down to a point" is a (formally) quite simple when working with factorization algebras.
//Remark:// This application of factorization algebras is due to Lurie \cite{cobord} and Morrison and Walker \cite{blob}. Our discussion here is modeled on Lurie's.
!! Topological factorization algebras
Let $M$ be a smooth $n$-manifold (possibly noncompact).
''Definition''
A factorization algebra $F$ is //topological// if it assigns the same value to every open set diffeomorphic to a $n$-ball.
Thanks to the gluing condition on a factorization algebra, knowing the value on all the balls in $M$ completely determines the factorization algebra. Since every $n$-manifold is covered by balls, a topological factorization algebra $F$ for one $n$-manifold induces a topological factorization algebra on //every// $n$-manifold! In other words, if we specify a topological factorization algebra on the open $n$-ball, we obtain a topological factorization algebra on all $n$-manifolds.
This extraordinary property of topological factorization algebras has its limitations, of course. In particular, the value we assign to a ball must be equipped the right structure. Unpacking the requirements, we discover that a topological factorization algebra on the open $n$-ball assigns an $E_n$-algebra to the open ball. But we want the factorization algebra to assign the same value to any open set diffeomorphic to the ball, and so we need this $E_n$-algebra to be invariant under the action of $Diff(B^n)$, the diffeomorphism group of the $n$-ball.
!! Constructing a TFT
Fix such an $E_n$ algebra $A$ in the category $C$. [NEED CONDITIONS HERE: MAYBE INFINIT~Y-CAT, DEFINITELY NEED (MOST) COLIMITS ...] From above we know how to associate a topological factorization algebra $A_M$ to every $n$-manifold $M$. But we can also construct a TFT $\mathcal{A}$ that extends down to a point as follows.
We begin by describing how $\mathcal{A}$ acts on the objects. Let $N$ be a codimension $k$ manifold. By crossing $N$ with the open $k$-ball, we obtain an $n$-manifold $\tilde{N} = N \times B^k$ and hence a topological factorization algebra $A_{\tilde{N}}$. Let $\mathcal{A}(N) = A_{\tilde{N}}(\tilde{N})$. Namely, the TFT $\mathcal{A}$ assigns to each manifold the global sections of the topological factorization algebra on the manifold (after it's thickened to an $n$-manifold).
Let $M$ be a manifold with boundary (of dimension at most $n$) and $N$ a connected component of that boundary. By taking a collared neighborhood of $N$ in $M$, we obtain a bijection between $M$ and $(M - N) \sqcup N \times [0,1]$. Thus we get an open embedding of $(~M-N) \sqcup N \times (0,1)$ into $M$. Applying our TFT procedure above, we see that we get a map
$\mathcal{A}(~M-N) \otimes \mathcal{A}(N) \rightarrow \mathcal{A}(M)$.
We can extend the action of $\mathcal{A}(N)$ on $\mathcal{A}(~M-N)$ to an action on all of $\mathcal{A}(M)$. [HOW???] Using this kind of construction, we see that a cobordism between two manifolds leads to a a bimodule over the values associated to each boundary component.
WHAT I'M WRITING IS TOO SIMILAR TO JACOB'S
/***
Description: Contains the stuff you need to use Tiddlyspot
Note, you also need UploadPlugin, PasswordOptionPlugin and LoadRemoteFileThroughProxy
from http://tiddlywiki.bidix.info for a complete working Tiddlyspot site.
***/
//{{{
// edit this if you are migrating sites or retrofitting an existing TW
config.tiddlyspotSiteId = 'factorization';
// make it so you can by default see edit controls via http
config.options.chkHttpReadOnly = false;
window.readOnly = false; // make sure of it (for tw 2.2)
window.showBackstage = true; // show backstage too
// disable autosave in d3
if (window.location.protocol != "file:")
config.options.chkGTDLazyAutoSave = false;
// tweak shadow tiddlers to add upload button, password entry box etc
with (config.shadowTiddlers) {
SiteUrl = 'http://'+config.tiddlyspotSiteId+'.tiddlyspot.com';
SideBarOptions = SideBarOptions.replace(/(<<saveChanges>>)/,"$1<<tiddler TspotSidebar>>");
OptionsPanel = OptionsPanel.replace(/^/,"<<tiddler TspotOptions>>");
DefaultTiddlers = DefaultTiddlers.replace(/^/,"[[WelcomeToTiddlyspot]] ");
MainMenu = MainMenu.replace(/^/,"[[WelcomeToTiddlyspot]] ");
}
// create some shadow tiddler content
merge(config.shadowTiddlers,{
'TspotOptions':[
"tiddlyspot password:",
"<<option pasUploadPassword>>",
""
].join("\n"),
'TspotControls':[
"| tiddlyspot password:|<<option pasUploadPassword>>|",
"| site management:|<<upload http://" + config.tiddlyspotSiteId + ".tiddlyspot.com/store.cgi index.html . . " + config.tiddlyspotSiteId + ">>//(requires tiddlyspot password)//<br>[[control panel|http://" + config.tiddlyspotSiteId + ".tiddlyspot.com/controlpanel]], [[download (go offline)|http://" + config.tiddlyspotSiteId + ".tiddlyspot.com/download]]|",
"| links:|[[tiddlyspot.com|http://tiddlyspot.com/]], [[FAQs|http://faq.tiddlyspot.com/]], [[blog|http://tiddlyspot.blogspot.com/]], email [[support|mailto:support@tiddlyspot.com]] & [[feedback|mailto:feedback@tiddlyspot.com]], [[donate|http://tiddlyspot.com/?page=donate]]|"
].join("\n"),
'WelcomeToTiddlyspot':[
"This document is a ~TiddlyWiki from tiddlyspot.com. A ~TiddlyWiki is an electronic notebook that is great for managing todo lists, personal information, and all sorts of things.",
"",
"@@font-weight:bold;font-size:1.3em;color:#444; //What now?// @@ Before you can save any changes, you need to enter your password in the form below. Then configure privacy and other site settings at your [[control panel|http://" + config.tiddlyspotSiteId + ".tiddlyspot.com/controlpanel]] (your control panel username is //" + config.tiddlyspotSiteId + "//).",
"<<tiddler TspotControls>>",
"See also GettingStarted.",
"",
"@@font-weight:bold;font-size:1.3em;color:#444; //Working online// @@ You can edit this ~TiddlyWiki right now, and save your changes using the \"save to web\" button in the column on the right.",
"",
"@@font-weight:bold;font-size:1.3em;color:#444; //Working offline// @@ A fully functioning copy of this ~TiddlyWiki can be saved onto your hard drive or USB stick. You can make changes and save them locally without being connected to the Internet. When you're ready to sync up again, just click \"upload\" and your ~TiddlyWiki will be saved back to tiddlyspot.com.",
"",
"@@font-weight:bold;font-size:1.3em;color:#444; //Help!// @@ Find out more about ~TiddlyWiki at [[TiddlyWiki.com|http://tiddlywiki.com]]. Also visit [[TiddlyWiki.org|http://tiddlywiki.org]] for documentation on learning and using ~TiddlyWiki. New users are especially welcome on the [[TiddlyWiki mailing list|http://groups.google.com/group/TiddlyWiki]], which is an excellent place to ask questions and get help. If you have a tiddlyspot related problem email [[tiddlyspot support|mailto:support@tiddlyspot.com]].",
"",
"@@font-weight:bold;font-size:1.3em;color:#444; //Enjoy :)// @@ We hope you like using your tiddlyspot.com site. Please email [[feedback@tiddlyspot.com|mailto:feedback@tiddlyspot.com]] with any comments or suggestions."
].join("\n"),
'TspotSidebar':[
"<<upload http://" + config.tiddlyspotSiteId + ".tiddlyspot.com/store.cgi index.html . . " + config.tiddlyspotSiteId + ">><html><a href='http://" + config.tiddlyspotSiteId + ".tiddlyspot.com/download' class='button'>download</a></html>"
].join("\n")
});
//}}}
| !date | !user | !location | !storeUrl | !uploadDir | !toFilename | !backupdir | !origin |
| 26/07/2011 18:17:54 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . | ok |
| 26/07/2011 18:31:00 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . | ok |
| 27/07/2011 15:35:07 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . | ok |
| 27/07/2011 18:33:12 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . | ok |
| 27/07/2011 19:13:33 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . |
| 29/07/2011 11:16:40 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . |
| 19/09/2011 16:47:05 | Owen | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . |
| 23/10/2011 14:18:42 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . |
| 07/12/2011 17:30:50 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . | ok |
| 07/12/2011 17:42:35 | Kevin | [[/|http://factorization.tiddlyspot.com/]] | [[store.cgi|http://factorization.tiddlyspot.com/store.cgi]] | . | [[index.html | http://factorization.tiddlyspot.com/index.html]] | . |
/***
|''Name:''|UploadPlugin|
|''Description:''|Save to web a TiddlyWiki|
|''Version:''|4.1.3|
|''Date:''|Feb 24, 2008|
|''Source:''|http://tiddlywiki.bidix.info/#UploadPlugin|
|''Documentation:''|http://tiddlywiki.bidix.info/#UploadPluginDoc|
|''Author:''|BidiX (BidiX (at) bidix (dot) info)|
|''License:''|[[BSD open source license|http://tiddlywiki.bidix.info/#%5B%5BBSD%20open%20source%20license%5D%5D ]]|
|''~CoreVersion:''|2.2.0|
|''Requires:''|PasswordOptionPlugin|
***/
//{{{
version.extensions.UploadPlugin = {
major: 4, minor: 1, revision: 3,
date: new Date("Feb 24, 2008"),
source: 'http://tiddlywiki.bidix.info/#UploadPlugin',
author: 'BidiX (BidiX (at) bidix (dot) info',
coreVersion: '2.2.0'
};
//
// Environment
//
if (!window.bidix) window.bidix = {}; // bidix namespace
bidix.debugMode = false; // true to activate both in Plugin and UploadService
//
// Upload Macro
//
config.macros.upload = {
// default values
defaultBackupDir: '', //no backup
defaultStoreScript: "store.php",
defaultToFilename: "index.html",
defaultUploadDir: ".",
authenticateUser: true // UploadService Authenticate User
};
config.macros.upload.label = {
promptOption: "Save and Upload this TiddlyWiki with UploadOptions",
promptParamMacro: "Save and Upload this TiddlyWiki in %0",
saveLabel: "save to web",
saveToDisk: "save to disk",
uploadLabel: "upload"
};
config.macros.upload.messages = {
noStoreUrl: "No store URL in parmeters or options",
usernameOrPasswordMissing: "Username or password missing"
};
config.macros.upload.handler = function(place,macroName,params) {
if (readOnly)
return;
var label;
if (document.location.toString().substr(0,4) == "http")
label = this.label.saveLabel;
else
label = this.label.uploadLabel;
var prompt;
if (params[0]) {
prompt = this.label.promptParamMacro.toString().format([this.destFile(params[0],
(params[1] ? params[1]:bidix.basename(window.location.toString())), params[3])]);
} else {
prompt = this.label.promptOption;
}
createTiddlyButton(place, label, prompt, function() {config.macros.upload.action(params);}, null, null, this.accessKey);
};
config.macros.upload.action = function(params)
{
// for missing macro parameter set value from options
if (!params) params = {};
var storeUrl = params[0] ? params[0] : config.options.txtUploadStoreUrl;
var toFilename = params[1] ? params[1] : config.options.txtUploadFilename;
var backupDir = params[2] ? params[2] : config.options.txtUploadBackupDir;
var uploadDir = params[3] ? params[3] : config.options.txtUploadDir;
var username = params[4] ? params[4] : config.options.txtUploadUserName;
var password = config.options.pasUploadPassword; // for security reason no password as macro parameter
// for still missing parameter set default value
if ((!storeUrl) && (document.location.toString().substr(0,4) == "http"))
storeUrl = bidix.dirname(document.location.toString())+'/'+config.macros.upload.defaultStoreScript;
if (storeUrl.substr(0,4) != "http")
storeUrl = bidix.dirname(document.location.toString()) +'/'+ storeUrl;
if (!toFilename)
toFilename = bidix.basename(window.location.toString());
if (!toFilename)
toFilename = config.macros.upload.defaultToFilename;
if (!uploadDir)
uploadDir = config.macros.upload.defaultUploadDir;
if (!backupDir)
backupDir = config.macros.upload.defaultBackupDir;
// report error if still missing
if (!storeUrl) {
alert(config.macros.upload.messages.noStoreUrl);
clearMessage();
return false;
}
if (config.macros.upload.authenticateUser && (!username || !password)) {
alert(config.macros.upload.messages.usernameOrPasswordMissing);
clearMessage();
return false;
}
bidix.upload.uploadChanges(false,null,storeUrl, toFilename, uploadDir, backupDir, username, password);
return false;
};
config.macros.upload.destFile = function(storeUrl, toFilename, uploadDir)
{
if (!storeUrl)
return null;
var dest = bidix.dirname(storeUrl);
if (uploadDir && uploadDir != '.')
dest = dest + '/' + uploadDir;
dest = dest + '/' + toFilename;
return dest;
};
//
// uploadOptions Macro
//
config.macros.uploadOptions = {
handler: function(place,macroName,params) {
var wizard = new Wizard();
wizard.createWizard(place,this.wizardTitle);
wizard.addStep(this.step1Title,this.step1Html);
var markList = wizard.getElement("markList");
var listWrapper = document.createElement("div");
markList.parentNode.insertBefore(listWrapper,markList);
wizard.setValue("listWrapper",listWrapper);
this.refreshOptions(listWrapper,false);
var uploadCaption;
if (document.location.toString().substr(0,4) == "http")
uploadCaption = config.macros.upload.label.saveLabel;
else
uploadCaption = config.macros.upload.label.uploadLabel;
wizard.setButtons([
{caption: uploadCaption, tooltip: config.macros.upload.label.promptOption,
onClick: config.macros.upload.action},
{caption: this.cancelButton, tooltip: this.cancelButtonPrompt, onClick: this.onCancel}
]);
},
options: [
"txtUploadUserName",
"pasUploadPassword",
"txtUploadStoreUrl",
"txtUploadDir",
"txtUploadFilename",
"txtUploadBackupDir",
"chkUploadLog",
"txtUploadLogMaxLine"
],
refreshOptions: function(listWrapper) {
var opts = [];
for(i=0; i<this.options.length; i++) {
var opt = {};
opts.push();
opt.option = "";
n = this.options[i];
opt.name = n;
opt.lowlight = !config.optionsDesc[n];
opt.description = opt.lowlight ? this.unknownDescription : config.optionsDesc[n];
opts.push(opt);
}
var listview = ListView.create(listWrapper,opts,this.listViewTemplate);
for(n=0; n<opts.length; n++) {
var type = opts[n].name.substr(0,3);
var h = config.macros.option.types[type];
if (h && h.create) {
h.create(opts[n].colElements['option'],type,opts[n].name,opts[n].name,"no");
}
}
},
onCancel: function(e)
{
backstage.switchTab(null);
return false;
},
wizardTitle: "Upload with options",
step1Title: "These options are saved in cookies in your browser",
step1Html: "<input type='hidden' name='markList'></input><br>",
cancelButton: "Cancel",
cancelButtonPrompt: "Cancel prompt",
listViewTemplate: {
columns: [
{name: 'Description', field: 'description', title: "Description", type: 'WikiText'},
{name: 'Option', field: 'option', title: "Option", type: 'String'},
{name: 'Name', field: 'name', title: "Name", type: 'String'}
],
rowClasses: [
{className: 'lowlight', field: 'lowlight'}
]}
};
//
// upload functions
//
if (!bidix.upload) bidix.upload = {};
if (!bidix.upload.messages) bidix.upload.messages = {
//from saving
invalidFileError: "The original file '%0' does not appear to be a valid TiddlyWiki",
backupSaved: "Backup saved",
backupFailed: "Failed to upload backup file",
rssSaved: "RSS feed uploaded",
rssFailed: "Failed to upload RSS feed file",
emptySaved: "Empty template uploaded",
emptyFailed: "Failed to upload empty template file",
mainSaved: "Main TiddlyWiki file uploaded",
mainFailed: "Failed to upload main TiddlyWiki file. Your changes have not been saved",
//specific upload
loadOriginalHttpPostError: "Can't get original file",
aboutToSaveOnHttpPost: 'About to upload on %0 ...',
storePhpNotFound: "The store script '%0' was not found."
};
bidix.upload.uploadChanges = function(onlyIfDirty,tiddlers,storeUrl,toFilename,uploadDir,backupDir,username,password)
{
var callback = function(status,uploadParams,original,url,xhr) {
if (!status) {
displayMessage(bidix.upload.messages.loadOriginalHttpPostError);
return;
}
if (bidix.debugMode)
alert(original.substr(0,500)+"\n...");
// Locate the storeArea div's
var posDiv = locateStoreArea(original);
if((posDiv[0] == -1) || (posDiv[1] == -1)) {
alert(config.messages.invalidFileError.format([localPath]));
return;
}
bidix.upload.uploadRss(uploadParams,original,posDiv);
};
if(onlyIfDirty && !store.isDirty())
return;
clearMessage();
// save on localdisk ?
if (document.location.toString().substr(0,4) == "file") {
var path = document.location.toString();
var localPath = getLocalPath(path);
saveChanges();
}
// get original
var uploadParams = new Array(storeUrl,toFilename,uploadDir,backupDir,username,password);
var originalPath = document.location.toString();
// If url is a directory : add index.html
if (originalPath.charAt(originalPath.length-1) == "/")
originalPath = originalPath + "index.html";
var dest = config.macros.upload.destFile(storeUrl,toFilename,uploadDir);
var log = new bidix.UploadLog();
log.startUpload(storeUrl, dest, uploadDir, backupDir);
displayMessage(bidix.upload.messages.aboutToSaveOnHttpPost.format([dest]));
if (bidix.debugMode)
alert("about to execute Http - GET on "+originalPath);
var r = doHttp("GET",originalPath,null,null,username,password,callback,uploadParams,null);
if (typeof r == "string")
displayMessage(r);
return r;
};
bidix.upload.uploadRss = function(uploadParams,original,posDiv)
{
var callback = function(status,params,responseText,url,xhr) {
if(status) {
var destfile = responseText.substring(responseText.indexOf("destfile:")+9,responseText.indexOf("\n", responseText.indexOf("destfile:")));
displayMessage(bidix.upload.messages.rssSaved,bidix.dirname(url)+'/'+destfile);
bidix.upload.uploadMain(params[0],params[1],params[2]);
} else {
displayMessage(bidix.upload.messages.rssFailed);
}
};
// do uploadRss
if(config.options.chkGenerateAnRssFeed) {
var rssPath = uploadParams[1].substr(0,uploadParams[1].lastIndexOf(".")) + ".xml";
var rssUploadParams = new Array(uploadParams[0],rssPath,uploadParams[2],'',uploadParams[4],uploadParams[5]);
var rssString = generateRss();
// no UnicodeToUTF8 conversion needed when location is "file" !!!
if (document.location.toString().substr(0,4) != "file")
rssString = convertUnicodeToUTF8(rssString);
bidix.upload.httpUpload(rssUploadParams,rssString,callback,Array(uploadParams,original,posDiv));
} else {
bidix.upload.uploadMain(uploadParams,original,posDiv);
}
};
bidix.upload.uploadMain = function(uploadParams,original,posDiv)
{
var callback = function(status,params,responseText,url,xhr) {
var log = new bidix.UploadLog();
if(status) {
// if backupDir specified
if ((params[3]) && (responseText.indexOf("backupfile:") > -1)) {
var backupfile = responseText.substring(responseText.indexOf("backupfile:")+11,responseText.indexOf("\n", responseText.indexOf("backupfile:")));
displayMessage(bidix.upload.messages.backupSaved,bidix.dirname(url)+'/'+backupfile);
}
var destfile = responseText.substring(responseText.indexOf("destfile:")+9,responseText.indexOf("\n", responseText.indexOf("destfile:")));
displayMessage(bidix.upload.messages.mainSaved,bidix.dirname(url)+'/'+destfile);
store.setDirty(false);
log.endUpload("ok");
} else {
alert(bidix.upload.messages.mainFailed);
displayMessage(bidix.upload.messages.mainFailed);
log.endUpload("failed");
}
};
// do uploadMain
var revised = bidix.upload.updateOriginal(original,posDiv);
bidix.upload.httpUpload(uploadParams,revised,callback,uploadParams);
};
bidix.upload.httpUpload = function(uploadParams,data,callback,params)
{
var localCallback = function(status,params,responseText,url,xhr) {
url = (url.indexOf("nocache=") < 0 ? url : url.substring(0,url.indexOf("nocache=")-1));
if (xhr.status == 404)
alert(bidix.upload.messages.storePhpNotFound.format([url]));
if ((bidix.debugMode) || (responseText.indexOf("Debug mode") >= 0 )) {
alert(responseText);
if (responseText.indexOf("Debug mode") >= 0 )
responseText = responseText.substring(responseText.indexOf("\n\n")+2);
} else if (responseText.charAt(0) != '0')
alert(responseText);
if (responseText.charAt(0) != '0')
status = null;
callback(status,params,responseText,url,xhr);
};
// do httpUpload
var boundary = "---------------------------"+"AaB03x";
var uploadFormName = "UploadPlugin";
// compose headers data
var sheader = "";
sheader += "--" + boundary + "\r\nContent-disposition: form-data; name=\"";
sheader += uploadFormName +"\"\r\n\r\n";
sheader += "backupDir="+uploadParams[3] +
";user=" + uploadParams[4] +
";password=" + uploadParams[5] +
";uploaddir=" + uploadParams[2];
if (bidix.debugMode)
sheader += ";debug=1";
sheader += ";;\r\n";
sheader += "\r\n" + "--" + boundary + "\r\n";
sheader += "Content-disposition: form-data; name=\"userfile\"; filename=\""+uploadParams[1]+"\"\r\n";
sheader += "Content-Type: text/html;charset=UTF-8" + "\r\n";
sheader += "Content-Length: " + data.length + "\r\n\r\n";
// compose trailer data
var strailer = new String();
strailer = "\r\n--" + boundary + "--\r\n";
data = sheader + data + strailer;
if (bidix.debugMode) alert("about to execute Http - POST on "+uploadParams[0]+"\n with \n"+data.substr(0,500)+ " ... ");
var r = doHttp("POST",uploadParams[0],data,"multipart/form-data; ;charset=UTF-8; boundary="+boundary,uploadParams[4],uploadParams[5],localCallback,params,null);
if (typeof r == "string")
displayMessage(r);
return r;
};
// same as Saving's updateOriginal but without convertUnicodeToUTF8 calls
bidix.upload.updateOriginal = function(original, posDiv)
{
if (!posDiv)
posDiv = locateStoreArea(original);
if((posDiv[0] == -1) || (posDiv[1] == -1)) {
alert(config.messages.invalidFileError.format([localPath]));
return;
}
var revised = original.substr(0,posDiv[0] + startSaveArea.length) + "\n" +
store.allTiddlersAsHtml() + "\n" +
original.substr(posDiv[1]);
var newSiteTitle = getPageTitle().htmlEncode();
revised = revised.replaceChunk("<title"+">","</title"+">"," " + newSiteTitle + " ");
revised = updateMarkupBlock(revised,"PRE-HEAD","MarkupPreHead");
revised = updateMarkupBlock(revised,"POST-HEAD","MarkupPostHead");
revised = updateMarkupBlock(revised,"PRE-BODY","MarkupPreBody");
revised = updateMarkupBlock(revised,"POST-SCRIPT","MarkupPostBody");
return revised;
};
//
// UploadLog
//
// config.options.chkUploadLog :
// false : no logging
// true : logging
// config.options.txtUploadLogMaxLine :
// -1 : no limit
// 0 : no Log lines but UploadLog is still in place
// n : the last n lines are only kept
// NaN : no limit (-1)
bidix.UploadLog = function() {
if (!config.options.chkUploadLog)
return; // this.tiddler = null
this.tiddler = store.getTiddler("UploadLog");
if (!this.tiddler) {
this.tiddler = new Tiddler();
this.tiddler.title = "UploadLog";
this.tiddler.text = "| !date | !user | !location | !storeUrl | !uploadDir | !toFilename | !backupdir | !origin |";
this.tiddler.created = new Date();
this.tiddler.modifier = config.options.txtUserName;
this.tiddler.modified = new Date();
store.addTiddler(this.tiddler);
}
return this;
};
bidix.UploadLog.prototype.addText = function(text) {
if (!this.tiddler)
return;
// retrieve maxLine when we need it
var maxLine = parseInt(config.options.txtUploadLogMaxLine,10);
if (isNaN(maxLine))
maxLine = -1;
// add text
if (maxLine != 0)
this.tiddler.text = this.tiddler.text + text;
// Trunck to maxLine
if (maxLine >= 0) {
var textArray = this.tiddler.text.split('\n');
if (textArray.length > maxLine + 1)
textArray.splice(1,textArray.length-1-maxLine);
this.tiddler.text = textArray.join('\n');
}
// update tiddler fields
this.tiddler.modifier = config.options.txtUserName;
this.tiddler.modified = new Date();
store.addTiddler(this.tiddler);
// refresh and notifiy for immediate update
story.refreshTiddler(this.tiddler.title);
store.notify(this.tiddler.title, true);
};
bidix.UploadLog.prototype.startUpload = function(storeUrl, toFilename, uploadDir, backupDir) {
if (!this.tiddler)
return;
var now = new Date();
var text = "\n| ";
var filename = bidix.basename(document.location.toString());
if (!filename) filename = '/';
text += now.formatString("0DD/0MM/YYYY 0hh:0mm:0ss") +" | ";
text += config.options.txtUserName + " | ";
text += "[["+filename+"|"+location + "]] |";
text += " [[" + bidix.basename(storeUrl) + "|" + storeUrl + "]] | ";
text += uploadDir + " | ";
text += "[[" + bidix.basename(toFilename) + " | " +toFilename + "]] | ";
text += backupDir + " |";
this.addText(text);
};
bidix.UploadLog.prototype.endUpload = function(status) {
if (!this.tiddler)
return;
this.addText(" "+status+" |");
};
//
// Utilities
//
bidix.checkPlugin = function(plugin, major, minor, revision) {
var ext = version.extensions[plugin];
if (!
(ext &&
((ext.major > major) ||
((ext.major == major) && (ext.minor > minor)) ||
((ext.major == major) && (ext.minor == minor) && (ext.revision >= revision))))) {
// write error in PluginManager
if (pluginInfo)
pluginInfo.log.push("Requires " + plugin + " " + major + "." + minor + "." + revision);
eval(plugin); // generate an error : "Error: ReferenceError: xxxx is not defined"
}
};
bidix.dirname = function(filePath) {
if (!filePath)
return;
var lastpos;
if ((lastpos = filePath.lastIndexOf("/")) != -1) {
return filePath.substring(0, lastpos);
} else {
return filePath.substring(0, filePath.lastIndexOf("\\"));
}
};
bidix.basename = function(filePath) {
if (!filePath)
return;
var lastpos;
if ((lastpos = filePath.lastIndexOf("#")) != -1)
filePath = filePath.substring(0, lastpos);
if ((lastpos = filePath.lastIndexOf("/")) != -1) {
return filePath.substring(lastpos + 1);
} else
return filePath.substring(filePath.lastIndexOf("\\")+1);
};
bidix.initOption = function(name,value) {
if (!config.options[name])
config.options[name] = value;
};
//
// Initializations
//
// require PasswordOptionPlugin 1.0.1 or better
bidix.checkPlugin("PasswordOptionPlugin", 1, 0, 1);
// styleSheet
setStylesheet('.txtUploadStoreUrl, .txtUploadBackupDir, .txtUploadDir {width: 22em;}',"uploadPluginStyles");
//optionsDesc
merge(config.optionsDesc,{
txtUploadStoreUrl: "Url of the UploadService script (default: store.php)",
txtUploadFilename: "Filename of the uploaded file (default: in index.html)",
txtUploadDir: "Relative Directory where to store the file (default: . (downloadService directory))",
txtUploadBackupDir: "Relative Directory where to backup the file. If empty no backup. (default: ''(empty))",
txtUploadUserName: "Upload Username",
pasUploadPassword: "Upload Password",
chkUploadLog: "do Logging in UploadLog (default: true)",
txtUploadLogMaxLine: "Maximum of lines in UploadLog (default: 10)"
});
// Options Initializations
bidix.initOption('txtUploadStoreUrl','');
bidix.initOption('txtUploadFilename','');
bidix.initOption('txtUploadDir','');
bidix.initOption('txtUploadBackupDir','');
bidix.initOption('txtUploadUserName','');
bidix.initOption('pasUploadPassword','');
bidix.initOption('chkUploadLog',true);
bidix.initOption('txtUploadLogMaxLine','10');
// Backstage
merge(config.tasks,{
uploadOptions: {text: "upload", tooltip: "Change UploadOptions and Upload", content: '<<uploadOptions>>'}
});
config.backstageTasks.push("uploadOptions");
//}}}
Cumrun Vafa and Edward Witten //A strong coupling test of $S$-duality//, 1994
<!--{{{-->
<div class='toolbar' macro='toolbar [[ToolbarCommands::ViewToolbar]]'></div>
<div class='title' macro='view title'></div>
<div class='subtitle'><span macro='view modifier link'></span>, <span macro='view modified date'></span> (<span macro='message views.wikified.createdPrompt'></span> <span macro='view created date'></span>)</div>
<div class='tagging' macro='tagging'></div>
<div class='tagged' macro='tags'></div>
<div class='viewer' macro='view text wikified'></div>
<div class='story'>
<div macro='storyViewer table_of_contents list prompt:"Table of contents"'></div>
</div>
<!--<div class = 'viewer' macro='tabs txtDiscussionTab
Page Page CurrentTiddler Discussion Discussion DiscussionTiddler'> (remove previous line and uncomment this for discussion tabs) </div>-->
<!--}}}-->
Edward Witten, //Two-dimensional models with (0,2) supersymmetry: perturbative aspects// hep-th/0504078
config.macros.discussion.titlefmt="_%who%_%UTC%";
config.macros.discussion.tags="comment";
//{{{
jsMath.Extension.Require("AMSmath");
jsMath.Extension.Require("AMSsymbols");
jsMath.Extension.Require("newcommand");
//defining fonts
jsMath.Font.Load("eufm10");
jsMath.Macro('mathfrak','{\\eufm{ #1 }}',1)
jsMath.Font.Load("rsfs10");
jsMath.Macro('mathscr','{\\rsfs {#1}}',1)
jsMath.Font.Load('bbold10');
jsMath.Macro('mathbb','{\\bbold {#1}}',1)
//macros taking arguments
jsMath.Macro('op','{\\mathop{\\rm #1}}',1);
jsMath.Macro('operatorname','{\\mathop{\\rm #1}}',1);
jsMath.Macro('mf','{\\mathfrak {#1} }',1);
jsMath.Macro('mscr','{\\mathscr {#1}} ',1);
jsMath.Macro('mbf','\\mathbf {#1} ',1);
jsMath.Macro('mbb','\\mathbb {#1} ',1);
jsMath.Macro('mc','\\mathcal {#1} ',1);
jsMath.Macro('norm','\\left\\| #1 \\right\\|',1);
jsMath.Macro('abs','\\left| #1 \\right|',1);
jsMath.Macro('ip','\\left\\langle #1 \\right\\rangle',1);
jsMath.Macro('br','\\overline{#1}',1);
jsMath.Macro('til','\\widetilde{#1}',1);
jsMath.Macro('wtil','\\widetilde{#1}',1);
jsMath.Macro('what','\\widehat{#1}',1);
jsMath.Macro('xto','\\xrightarrow{#1}',1);
//macros for symbols, etc.
jsMath.Macro('Lap','\\op{D}');
jsMath.Macro('R','{\\mathbb R}');
jsMath.Macro('Q','{\\mathbb Q}');
jsMath.Macro('N','{\\mathbb N}');
jsMath.Macro('Z','{\\mathbb Z}');
jsMath.Macro('C','{\\mathbb C}');
jsMath.Macro('Oo','{\\mathscr {O}}');
jsMath.Macro('Ool','{\\mathscr {O}_l}');
jsMath.Macro('E','{\\mathscr {E}}');
jsMath.Macro('L','{\\mathscr {L}}');
jsMath.Macro('sC','{\\mathscr {C}}');
jsMath.Macro('sL','{\\mathscr {L}}');
jsMath.Macro('A','\\mathscr A');
jsMath.Macro('K','\\mathbb K');
jsMath.Macro('F','\\mathcal F');
jsMath.Macro('eps','\\epsilon');
jsMath.Macro('g','\\mathfrak g');
jsMath.Macro('gl','\\mathfrak gl');
jsMath.Macro('iso','\\cong');
jsMath.Macro('d','\\mathrm d');
jsMath.Macro('GF','Q^{GF}');
jsMath.Macro('into','\\hookrightarrow');
jsMath.Macro('det','\\op{det}');
jsMath.Macro('Hom','\\op{Hom}');
jsMath.Macro('End','\\op{End}');
jsMath.Macro('Vect','\\op{Vect}');
jsMath.Macro('Comp','\\op{Comp}');
jsMath.Macro('Top','\\op{Top}');
jsMath.Macro('Aut','\\op{Aut}');
jsMath.Macro('Sym','\\op{Sym}');
jsMath.Macro('csym','\\widehat{\Sym}');
jsMath.Macro('Diff','\\op{Diff}');
jsMath.Macro('cinfty','C^\\infty');
jsMath.Macro('Tr','\\op{Tr}');
jsMath.Macro('Ker','\\op{Ker}');
jsMath.Macro('Im','\\op{Im}');
jsMath.Macro('Ob','\\op{Ob}');
jsMath.Macro('Supp','\\op{Supp}');
jsMath.Macro('dbar','\\overline{\\partial}');
jsMath.Macro('PV','\\op{PV}');
jsMath.Macro('liminv', '\\varprojlim');
jsMath.Macro('limdir', '\\varinjlim');
jsMath.Macro('SP', '\\op{SP}');
jsMath.Macro('Res', '\\op{Res}');
jsMath.Macro('Fields', '\\op{Fields}');
jsMath.Macro('Obs', '\\op{Obs}');
jsMath.Macro('mod','\\,\\,\\op{mod}\\,\\,');
jsMath.Macro('EL','\\mathcal{EL}');
jsMath.Macro('Px','\\op{Param}');
jsMath.Macro('colim','\\operatorname{colim}');
jsMath.Macro('ObsStrict','\\op{Obs}');
jsMath.Macro('ObsHomotopy','\\op{Obs}_{h}');
jsMath.Macro('Gr','\\op{Gr}');
jsMath.Macro('defeq', '\\overset{\\text{def}}{=}');
//}}}
/***
|Name|LooseLinksPlugin|
|Source|http://www.TiddlyTools.com/#LooseLinksPlugin|
|Documentation|http://www.TiddlyTools.com/#LooseLinksPlugin|
|Version|1.1.2|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|case-folded/space-folded wiki words|
!!!!!Documentation
<<<
This plugin extends the TiddlyWiki core handling for tiddler links to permit use of non-WikiWord variations of mixed-case and/or added/omitted spaces within double-bracketed text with titles of //existing// tiddlers, using a 'loose' (case-folded/space-folded) comparison. This allows text that occurs in normal prose to be more easily linked to tiddler titles by using double-brackets without the full 'pretty link' syntax. For example:
{{{
[[CoreTweaks]], [[coreTweaks]], [[core tweaks]],
[[CORE TWEAKS]], [[CoRe TwEaKs]], [[coreTWEAKS]]
}}}
>[[CoreTweaks]], [[coreTweaks]], [[core tweaks]],
>[[CORE TWEAKS]], [[CoRe TwEaKs]], [[coreTWEAKS]]
<<<
!!!!!Configuration
<<<
<<option chkLooseLinks>> Allow case-folded and/or space-folded text to link to existing tiddler titles
"""<<option chkLooseLinks>>"""
<<<
!!!!!Revisions
<<<
2009.08.14 [1.1.2] corrected call to addNotification()
2009.08.14 [1.1.1] code cleanup
2009.08.02 [1.1.0] big performance rewrite: use cached LooseLinksMap[] instead of scanning each time
2009.01.06 [1.0.0] converted to stand-alone plugin
2008.10.14 [0.0.0] initial release (as [[CoreTweaks]] #664 - http://trac.tiddlywiki.org/ticket/664)
<<<
!!!!!Code
***/
//{{{
version.extensions.LooseLinksPlugin={major:1, minor:1, revision:2, date: new Date(2009,8,15)};
//if (!config.options.chkLooseLinks)
// config.options.chkLooseLinks=false; // default to standard
if (window.caseFold_createTiddlyLink===undefined) { // only once
window.caseFold_createTiddlyLink = window.createTiddlyLink;
window.createTiddlyLink = function(place,title,includeText,className) {
var btn=window.caseFold_createTiddlyLink.apply(this,arguments); // create core link
// if (!config.options.chkLooseLinks) return btn;
if (store.getTiddlerText(title)) return btn; // matching tiddler (or shadow) exists
var tid=window.getLooseLinksMap()[title.toLowerCase().replace(/\s/g,'')];
if (tid) {
var i=getTiddlyLinkInfo(tid,className);
btn.setAttribute('tiddlyLink',tid);
btn.title=i.subTitle;
btn.className=i.classes;
}
return btn;
}
}
window.getLooseLinksMap=function(title) {
// if (!config.options.chkLooseLinks) return {}; // disable
if (!config.looseLinksMap) { // init/cache on demand
config.looseLinksMap={};
store.forEachTiddler(function(title,tiddler){
config.looseLinksMap[title.toLowerCase().replace(/\s/g,'')]=title;
});
}
if (title) config.looseLinksMap[title.toLowerCase().replace(/\s/g,'')]=title; // update
return config.looseLinksMap;
}
store.addNotification(null,window.getLooseLinksMap); // notify
//}}}
/***
|Name|Plugin: arXiv Links|
|Created by|BobMcElrath|
|Email|my first name at my last name dot org|
|Location|http://bob.mcelrath.org/tiddlyjsmath-2.0.3.html|
|Version|1.0|
|Requires|[[TiddlyWiki|http://www.tiddlywiki.com]] ≥ 2.0.3|
!Description
This formatting plugin will render links to the [[arXiv|http://www.arxiv.org]] preprint system. If you type a paper reference such as hep-ph/0509024, it will be rendered as an external link to the abstract of that paper.
!Installation
Add this tiddler to your tiddlywiki, and give it the {{{systemConfig}}} tag.
!History
* 1-Feb-06, version 1.0, Initial release
!Code
***/
//{{{
config.formatters.push({
name: "arXivLinks",
match: "\\b(?:astro-ph|cond-mat|hep-ph|hep-th|hep-lat|gr-qc|nucl-ex|nucl-th|quant-ph|(?:cs|math|nlin|physics|q-bio)(?:\\.[A-Z]{2})?)/[0-9]{7}\\b",
element: "a",
handler: function(w) {
var e = createExternalLink(w.output, "http://arxiv.org/abs/"+w.matchText);
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!! The sheaf of quantum field theories and the sheaf of observables
(This article is a stub, and just contains a badly written sketch of the statement and proof.)
In \cite{webbook} it was shown that ~QFTs form a sheaf on a manifold. Factorization algebras also form a sheaf; we want to show we have a map of sheaves.
The key step is to show that if $V \subset U \subset M$, then we can think of $\op{Obs}(V)$ in either the QFT restricted to $U$ or the original QFT on $M$, and we get canonically isomorphic spaces.
The proof of this is another trick using compact support of observables. (It might be convenient, in general, to write $\op{Obs}(U)$ as an inverse limit over indices $(i,k)$ of a direct limit over compact subsets of $U$; this would make writing things much easier in several places).
For observables supported on a compact subset of $V$ things are OK. As, we can take a cut-off function $f$ on $U$ and $g$ on $M$ which are the same on $K \times K$. Then, if $I$ is the theory on $M$ and $\til{I}$ is that on $U$, then $I[g]$ and $\til{I}[f]$ act the same way on functionals supported on $K$, pretty much. At least, here, the difference between them is smooth, so we can translate back and forth between them without much difficulty.
# [[ Introduction, overview, and physical motivation|Introduction]]
## [[ Factorization algebras in quantum mechanics|The motivating example of quantum mechanics]]
## [[A preliminary definition of prefactorization algebras]]
## [[Prefactorization algebras in quantum field theory]]
# The Main Theorems and a Guide to the Paper
## [[Classical field theory and factorization algebras]]
## [[Quantum field theory and factorization algebras]]
## [[The weak quantization theorem]]
## [[The strong quantization theorem]]
# Factorization algebras and basic examples
## Definition of a [[prefactorization algebra |Prefactorization algebras]]
## [[Concrete examples of prefactorization algebras]]
## Definition of a [[factorization algebra |Factorization algebra]]
## [[Factorization algebras from cosheaves]]
## [[Locally constant factorization algebras]] and $E_n$ algebras
## [[Ordinary quantum mechanics as a factorization algebra]]
# Properties of the category of factorization algebras
## [[The category of factorization algebras]]
## [[Pushforward]]
## [[Extension from a basis]]
## [[Pullback]]
## [[Descent]]
# Operads and factorization algebras
## [[Structured factorization algebras]]
## The [[P_0 operad]]
## The [[Beilinson-Drinfeld operad]]
## [[Lax algebras over an operad]]
# Classical field theory
## [[Motivational overview|Introduction to classical field theory]]
## [[Elliptic moduli problems and local Lie algebras]]
### [[Formal moduli problems and Lie algebras]]
### [[Elliptic moduli problems]]
### [[Examples of elliptic moduli problems related to scalar field theories]]
### [[Examples of elliptic moduli problems related to gauge theories]]
### [[Cochains of a local Lie algebra]]
### [[D-modules and local Lie algebras]]
## Definition of a classical field theory
### [[The classical BV formalism in finite dimensions]]
### [[The classical BV formalism in infinite dimensions]]
### [[The exterior derivative of a local action functional]]
### [[Field theories from action functionals]]
### [[A succinct definition of a classical field theory|Definition of classical field theory]]
## Examples of field theories
### [[Examples of field theories from action functionals]]
### [[Cotangent field theories]]
## [[The graded Poisson structure on classical observables]]
### [[The Poisson structure for free field theories]]
### [[The Poisson structure for cotangent field theories]]
### [[The Poisson structure for a general classical field theory]]
## Symmetries and conserved quantities
### [[Symmetries of a classical field theory]]
### [[Symmetries and local functionals]]
##[[Symmetries and conserved quantities]]
# Quantum field theory
## [[Outline of the deformation quantization picture]]
## [[Global observables]]
## [[Informal description of the local observables]]
## [[Parametrices]]
## [[Effective interactions and the quantum master equation using parametrices]]
## [[Global observables using parametrices]]
## [[The factorization algebra of observables of a quantum field theory |Quantum observables]]
### [[Local quantum observables]]
### [[The prefactorization algebra of observables]]
### [[The factorization algebra of observables]]
## [[Correlation functions]]
## [[The sheaf of theories|sheaf-of-theories]]
# Examples of ~QFTs
## [[The Weyl algebra and the free field on the real line]]
## [[Free fields]]
### [[The Heisenberg algebra construction]]
### Free fields and [[determinants of complexes|Determinants of complexes]]
## [[The Weyl algebra redux]]
## [[The free holomorphic boson]]
# [[Overview of perturbative quantum field theory]]
## [[Local action functional]]
## [[Quantum field theory]]
## [[BV theory]]
## [[Local-effective correspondence]]
# [[Appendix on homological algebra and functional analysis]]
## [[Classes of sections of a vector bundle]]
## [[Classes of functions on the space of sections of a vector bundle]]
## [[Derivations]]
## [[Nuclear spaces|Overview of nuclear spaces]]
## The [[Completed inductive tensor product]]
## [[Topological cochain complexes]]
## An [[Atiyah-Bott lemma]] for elliptic complexes on open sets
# [[Appendix on dg manifolds and the BV formalism]]
## [[Signs and other conventions]]
## [[Kernels in the BV formalism]]
## [[Differential graded manifolds]]
## [[Derived critical locus]]
/%##[[The Euler-Lagrange equations|Euler-Lagrange]]
## [[Derived Euler-Lagrange equations]]
## [[Gauge theories and nonlinear sigma models in classical field theory]]
## [[Symplectic structures on field spaces]]
## [[ Formal definition|Definition of classical field theory]]
## [[Observables of a classical field theory]]%/