  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.
