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.

57 page, 5 figures. Final version, to appear in J. Top

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• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
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Cited by in corpus (41)

• Modular extensions of unitary braided fusion categories and 2+1D topological/SPT orders with symmetries
• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
• Gapless edges of 2d topological orders and enriched monoidal categories
• Gapped boundary theories in three dimensions
• On dualizability of braided tensor categories
• The center functor is fully faithful
• On the combinatorics of exact Lagrangian surfaces
• Q-system completion for C* 2-categories
• Excision of Skein Categories and Factorisation Homology
• Rigid and Separable Algebras in Fusion 2-Categories
• Generalized Springer Theory for D-modules on a Reductive Lie Algebra
• Topological orders and factorization homology
• Introductory lectures on topological quantum field theory
• Invariants of 4-manifolds from Khovanov-Rozansky link homology
• Quantum K-theoretic geometric Satake
• Categorification of algebraic quantum field theories
• Factorization Homology and 4D TQFT
• Algebras of quantum monodromy data and decorated character varieties
• Finite presentations for stated skein algebras and lattice gauge field theory
• Reduced Tensor Product on the Drinfeld Center
• The Elliptic Drinfeld Center of a Premodular Category
• The Harish-Chandra isomorphism for quantum GL_2
• On bicolimits of $C^{\ast }$-categories
• Kauffman Skein Algebras and Quantum Teichmüller Spaces via Factorisation Homology
• Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras
• Projective representations of mapping class groups in combinatorial quantization
• Branes and DAHA Representations
• Quantum decorated character stacks
• The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality
• Stated skein modules of 3-manifolds and TQFT
• A geometric approach to equivariant factorization homology and nonabelian Poincaré duality
• The wonderful compactification for quantum groups
• Categorical Center of Higher Genera and 4D Factorization Homology
• Deformation Quantization via Categorical Factorization Homology
• The Jordan--Chevalley decomposition for $G$-bundles on elliptic curves
• The ${\mathfrak{gl}}_2$-Skein Module of Lens Spaces via the Torus and Solid Torus
• $C^{\ast }$-categorical prefactorization algebras for superselection sectors and topological order
• Stated skein algebras and their representations
• Natural symmetries of secondary Hochschild homology
• Quantum Harish-Chandra isomorphism for the double affine Hecke algebra of $G{L}_n$
• Equivalence of field theories: Crane-Yetter and the shadow