English

Integrating quantum groups over surfaces

Quantum Algebra 2018-08-15 v5 Algebraic Geometry Representation Theory

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)(0,1,2)-dimensional part of Crane-Yetter-Kauffman 4D TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group Uq(g)U_q(\mathfrak g) we obtain in this way an aspect of topologically twisted 4-dimensional N=4{\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 GG-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 Uq(g)U_q(\mathfrak g), and from the punctured torus we recover the algebra of quantum differential operators associated to Uq(g)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 D\mathcal D-modules.

Keywords

Cite

@article{arxiv.1501.04652,
  title  = {Integrating quantum groups over surfaces},
  author = {David Ben-Zvi and Adrien Brochier and David Jordan},
  journal= {arXiv preprint arXiv:1501.04652},
  year   = {2018}
}

Comments

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

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