Integrating quantum groups over surfaces

David Ben-Zvi, Adrien Brochier, David Jordan

Research output: Contribution to journalArticlepeer-review


Braided tensor categories give rise to 4-dimensional topological field theories, extending constructions of Crane-Yetter-Kauffman in the case of modular categories. In this paper we apply the mechanism of factorization homology in order to construct and compute the category-valued invariants of surfaces which form the two-dimensional part of the theory. Starting from modules for the Drinfeld-Jimbo quantum group Uq(g) we obtain in this way a form of topologically twisted 4-dimensional 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 Uq(g), and from the punctured torus we recover the algebra of quantum differential operators associated to Uq(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-modules.
Original languageEnglish
Pages (from-to)873-916
Number of pages44
JournalJournal of Topology
Issue number4
Early online date3 Aug 2018
Publication statusPublished - Dec 2018

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