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III International Conference "Quantum Topology"
June 22, 2016 11:20–12:10, Moscow, Steklov Mathematical Institute

Quantum cluster algebras and character varieties of SL(2,R)-monodromy problem

L. O. Chekhov
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L. O. Chekhov
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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. 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. Moreover, using the explicit parameterization of monodromy elements we can evaluate the Poisson and quantum algebras of monodromy matrices generated by the Poisson and quantum algebras of $\lambda$-lengths and show that these algebras are quadratic quasi-Poisson, or quasi-quantum, algebras. These algebras are invariant w.r.t. mutations of cluster algebras, which correspond to MCG transformations, and can be therefore lifted from $T_{g,s,n}$ to the moduil space $M_{g,s,n}$. Complexifying the cluster variables we obtain the character variety of $SL(2,C)$-monodromy problem.
The talk is based on the joint works with with M. Mazzocco and V. Roubtsov [1, 2, 3].
  • L. Chekhov and M. Mazzocco, Colliding holes in Riemann surfaces and quantum cluster algebras. arXiv:1509.07044.
  • L. Chekhov, M. Mazzocco, and V. Roubtsov, Painlev╨╣ monodromy manifolds, decorated character varieties and cluster algebras. arXiv:1511.03851.
  • L. Chekhov, M. Mazzocco, and V. Roubtsov, Decorated character varieties of monodromy manifolds and quantum cluster algebras. in preparation.

Language: English

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