Quantum character varieties and braided module categories

TL;DR

This paper develops a categorical framework for quantum character varieties of surfaces with boundaries and marked points, using braided tensor categories, factorization homology, and quantum Hamiltonian reduction, with applications to quantum groups and DAHA representations.

Contribution

It introduces the concept of braided module categories and quantum moment maps, linking quantum character varieties to quantum Hamiltonian reduction and establishing connections with quantum groups and DAHA.

Findings

Quantum character varieties are computed via factorization homology.

Braided module categories are characterized by quantum moment maps.

Connections to quantum groups and DAHA representations are established.

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 $\mathcal A=\operatorname{Rep_q} G$ with the category $\mathcal D_q(G/G)-\operatorname{mod}$ 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 (DAHA) $\mathbb{SH}_{q,t}$.