Integrating quantum groups over surfaces

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.