# Quantum character varieties and braided module categories

Selecta Mathematica, Jul 2018

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 $${\mathbb {SH}}_{q,t}$$.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00029-018-0426-y.pdf

David Ben-Zvi, Adrien Brochier, David Jordan. Quantum character varieties and braided module categories, Selecta Mathematica, 2018, 1-38, DOI: 10.1007/s00029-018-0426-y