Integrating quantum groups over surfaces

TL;DR

This work develops factorization homology valued in braided tensor categories to produce category-valued 2D topological field theories from quantum groups. It proves that for any oriented surface S, the quantum character variety ∫_S Rep_q G is equivalent to modules over an internal End algebra A_S, with A_S presented combinatorially from gluing patterns and yielding known quantum algebras (O_q(G), D_q(G), Alekseev moduli algebras) as concrete instances. The framework simultaneously provides a geometric- and algebraic route to quantization of character varieties, mapping-class and braid-group actions, and a topological underpinning for Betti Langlands-inspired phenomena. It also develops the necessary higher-categorical machinery (Barr–Beck reconstruction, Deligne–Kelly tensor product, and monadicity) for factorization homology in the linear, abelian setting, and offers extensive examples and connections to four-dimensional TFTs and elliptic structures. Altogether, the paper integrates topological, algebraic, and geometric perspectives to unify and extend quantum group quantizations of surface character varieties within a functorial, locality-driven framework.

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.

Figures (5)

• Figure 1: The punctured surface $\Sigma(P)$ constructed from a gluing pattern $P$.
• Figure 2: In the linked case, we have $P(1,1',2,2') = (1,3,2,4)$.
• Figure 3: The annulus constructed from a gluing pattern yields $A_{Ann}\cong a_{Ann}\cong \widetilde{H}$. The pair of pants yields the braided tensor product $a_{Pan}\cong \widetilde{H} \widetilde{\otimes}\widetilde{H}$.
• Figure 4: The punctured torus constructed from a gluing pattern yields $A_{T^2\backslash D}\cong \mathcal{D}_H$.
• Figure 5: At left, the $r$-punctured, genus $g$ surface from a gluing pattern yields the iterated tensor product of $g$ copies of $\mathcal{D}_H$ and $r-1$ copies of $\widetilde{H}$. At right, the annulus with framing of index $k$ is obtained by a coiled gluing pattern with a single coiled handle.

Theorems & Definitions (110)

• Remark 1.1
• Remark 1.2: Derived version
• Remark 1.3: Skein Categories
• Remark 1.4
• Theorem 1.5: Theorem \ref{['thm:main']}
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Remark 2.2