Department of Mathematics

Geometry and Topology

  •  Leonid Chekhov, Steklov Mathematical Institute
  •  Quantum cluster algebras from geometry
  •  02/02/2017
  •  2:00 PM - 2:50 PM
  •  C304 Wells Hall

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 technique; 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. I also discuss the very recent results enabling constructing monodromy matrices of SL(2)-connections out of the corresponding cluster variables. The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov



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