## EVENT DETAILS AND ABSTRACT

**Geometry/Physics Seminar**
**Title:** Quantum cluster algebras from geometry

**Speaker:** Leonid Chekhov

**Speaker Info:** Steklov Institute, Moscow

**Brief Description:**

**Abstract:**

We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann
surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$
bordered cusps located on boundaries of holes uniformized by Poincare
with
the character variety of $SL(2,R)$-monodromy problem. The effective
combinatorial description uses the fat graph structures; observables
are
geodesic functions of closed curves and $\lambda$-lengths of paths
starting and terminating at bordered cusps decorated by horocycles.
Such
geometry stems from special "chewing gum" moves corresponding to
colliding
holes (or sides of the same hole) in a Riemann surface with holes. We
derive Poisson and quantum structures on sets of observables relating
them
to quantum cluster algebras of Berenstein and Zelevinsky. A seed of
the
corresponding quantum cluster algebra corresponds to the partition of
$\Sigma_{g,s,n}$ into ideal triangles, $\lambda$-lengths of their sides
are cluster variables constituting a seed of the algebra; their number
$6g-6+3s+2n$ (and, correspondingly, the seed dimension) coincides with
the
dimension of $SL(2,R)$-character variety given by
$[SL(2,R)]^{2g+s+n-2}/\prod_{i=1}^n B_i$,
where $B_i$ are Borel subgroups associated with bordered cusps.
The talk is based on the joint papers with with M.Mazzocco and
V.Roubtsov

**Date:** Tuesday, January 17, 2017

**Time:** 1:00pm

**Where:** Lunt 103

**Contact Person:** Ezra Getzler

**Contact email:**

**Contact Phone:**

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