January 27: Gus Schrader

Title: Decorated character varieties and their cluster structure via factorization homology

Abstract:

Character varieties for surfaces are affine, generally singular, Poisson varieties, for which quantizations were constructed by Fock-Rosly and Alekseev-Grosse-Schomerus in the 90's. Subsequently Fock and Goncharov introduced the notion of a decorated character variety corresponding to a marked surface, a rational Poisson variety with a Poisson projection to the original character variety of the unmarked surface. Moreover, the ring of functions on a decorated character variety carries the additional structure of a cluster algebra. In general, however, the decorated character variety is not affine, and quantizing it amounts to constructing a quantum analog of its category of quasicoherent sheaves. I'll explain an approach to constructing such a quantization using the stratified factorization homology developed by Ayala-Francis-Tanaka, and show how one can recover the combinatorics of cluster charts on the decorated character variety from this construction. Joint work with D. Jordan, I. Le and A. Shapiro.