Categorification of skein algebras
Organizers
Speaker
Dylan Allegretti
Time
Monday, April 28, 2025 3:30 PM - 4:30 PM
Venue
Shuangqing-B627
Abstract
The skein algebra of a surface is a noncommutative algebra that quantizes the SL(2,C)-character variety of the surface. It has been intensively studied in quantum topology for more than thirty years. In an influential paper from 2014, D. Thurston suggested that the skein algebra should have a natural categorification where the product in the algebra arises from a monoidal structure on a category. In this talk, I will describe such a categorification of the skein algebra of a genus zero surface with boundary. I will first review the construction of the variety of triples, a remarkable geometric object introduced by Braverman, Finkelberg, and Nakajima in their study of Coulomb branches of 3d N=4 gauge theories. I will then explain how the skein algebra arises as the Grothendieck ring of the bounded derived category of equivariant coherent sheaves on the variety of triples equipped with a natural monoidal structure. This talk is based on work with Hyun Kyu Kim and Peng Shan.