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 ∫ _T^2𝒜 . We initiate a theory of character sheaves for quantum groups by identifying the torus integral of 𝒜=Rep_qG with the category 𝒟_q(G/G) -mod of equivariant quantum 𝒟 -modules. When G=GL_n , we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra 𝕊ℍ_q,t .