Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants ∫ SA of a surface S, determined by the choice of a braided tensor category A , and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for A , and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided A -modules are objects of the torus category ∫ T²A . We initiate a theory of character sheaves for quantum groups by identifying the torus integral of A=RepqG with the category Dq(G/G) -mod of equivariant quantum D -modules. When G=GLₙ , we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra SHq,t .