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 -dimensional part of Crane-Yetter-Kauffman 4D TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group we obtain in this way an aspect of topologically twisted 4-dimensional 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 -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 , and from the punctured torus we recover the algebra of quantum differential operators associated to . 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 -modules.
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