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 {\em braided module category} for $\mathcal{A}$, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called {\em 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 $\mathcal{A}=\mathrm{Re}{\mathrm{p}}_{\mathrm{q}}G$ with the category ${\mathcal{D}}_q(G/G)-\mathrm{mod}$ of equivariant quantum $\mathcal{D}$-modules. When $G=G{L}_n$, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra (DAHA) ${\mathbb{SH}}_{q,t}$.

33 pages, 5 figures. Final version, to appear in Sel. Math. New Ser