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Integrating quantum groups over surfaces

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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.

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math.QA 1

years

2026 1

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UNVERDICTED 1

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Defects in skein theory and TQFT

math.QA · 2026-06-05 · unverdicted · novelty 7.0

Defines defect skein modules for 3-manifolds with line and point defects and proves they match state spaces of defect Reshetikhin-Turaev TQFT for semisimple data.

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  • Defects in skein theory and TQFT math.QA · 2026-06-05 · unverdicted · none · ref 2 · internal anchor

    Defines defect skein modules for 3-manifolds with line and point defects and proves they match state spaces of defect Reshetikhin-Turaev TQFT for semisimple data.