Integrating quantum groups over surfaces

TL;DR

This paper develops a topological quantum field theory framework using factorization homology to produce and analyze quantum group-related algebras and character varieties on surfaces, connecting to the geometric Langlands program.

Contribution

It introduces a new topological construction of quantum group algebras and character varieties via factorization homology, generalizing known 4D TFTs and providing topological insights into their symmetries.

Findings

Explicit categories quantize character varieties on surfaces.

Recovered key quantum group algebras like the reflection equation algebra.

Provided topological explanations for mapping class group symmetries.

Abstract

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the $(0,1,2)$-dimensional part of Crane-Yetter-Kauffman 4D TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group $U_q(\mathfrak g)$ we obtain in this way an aspect of topologically twisted 4-dimensional ${\mathcal N}=4$ super Yang-Mills theory, the setting introduced by Kapustin-Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of $G$-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to $U_q(\mathfrak g)$, and from the punctured torus we recover the algebra of quantum differential operators associated to $U_q(\mathfrak g)$. From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum $\mathcal D$-modules.