Quantum character varieties and braided module categories

Abstract

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants $\int _S{\mathcal {A}}$ of a surface S, determined by the choice of a braided tensor category ${\mathcal {A}}$, 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 ${\mathcal {A}}$, 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 ${\mathcal {A}}$-modules are objects of the torus category $\int _{T^2}{\mathcal {A}}$. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of ${\mathcal {A}}={\text {Rep}}_{q}G$ with the category ${\mathcal {D}}_q(G/G)$-mod of equivariant quantum ${\mathcal {D}}$-modules. When $G=GL_n$, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra ${\mathbb {SH}}_{q,t}$.