We apply the mechanism of factorization homology to construct and computecategory-valued two-dimensional topological field theories associated tobraided tensor categories, generalizing the $(0,1,2)$-dimensional part ofCrane-Yetter-Kauffman 4D TFTs associated to modular categories. Starting frommodules for the Drinfeld-Jimbo quantum group $U_q(\mathfrak g)$ we obtain inthis way an aspect of topologically twisted 4-dimensional ${\mathcal N}=4$super Yang-Mills theory, the setting introduced by Kapustin-Witten for thegeometric Langlands program. For punctured surfaces, in particular, we produce explicit categories whichquantize character varieties (moduli of $G$-local systems) on the surface;these give uniform constructions of a variety of well-known algebras in quantumgroup theory. From the annulus, we recover the reflection equation algebraassociated to $U_q(\mathfrak g)$, and from the punctured torus we recover thealgebra of quantum differential operators associated to $U_q(\mathfrak g)$.From an arbitrary surface we recover Alekseev's moduli algebras. Ourconstruction gives an intrinsically topological explanation for well-knownmapping class group symmetries and braid group actions associated to thesealgebras, in particular the elliptic modular symmetry (difference Fouriertransform) of quantum $\mathcal D$-modules.
展开▼
