Mar 23 at 17:30 in 507

Title: Chiralization of surface character varieties and their cluster structure

Abstract:  The goal of the talk is to describe a somewhat speculative 'chiral' analog of the following well-known story in quantum topology. The SL(2,C) character variety of a topological surface has a Poisson structure given by Goldman's bracket for traces of holonomies around loops. Fock and Goncharov constructed a combinatorial quantization of this Poisson structure using its enhancement to a cluster structure on a framed version of the moduli space, and this quantization turns out to be governed algebraically by the quantum group U_q(sl2). It appears that an analogous chiral quantization can be constructed using the notion of chiral cluster transformations developed by Bershtein, Bourgine, Feigin and Shiraishi, in which generators of a finite-rank quantum torus are promoted to vertex operators on a Fock space and mutations to fermionic or q-Liouville reflections. In the chiral setting, the OPE of parallel transport observables are now controlled by the quantum affine algebra U_q(\widehat{sl}_2). 

Based on joint work with M. Bershtein and A. Shapiro.