Quantum character varieties and braided module categories

David Ben-Zvi, Adrien Brochier, David Jordan

Research output: Contribution to journalArticlepeer-review

Abstract

We study a categorical invariant of surfaces associated to a braided tensor category A, the factorization homology ∫_SA or quantum character variety of S. In our previous paper arXiv:1501.04652 we introduced these invariants and computed them for a punctured surface S as a category of modules for a canonical and explicit algebra, quantizing functions on the classical character variety. In this paper we compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points.
We identify the algebraic data governing marked points and boundary components with the notion of a {\em braided module category} for 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 A-modules are objects of the torus integral ∫_{T^2} A. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of A=RepqG with the category Dq(G/G)-mod of equivariant quantum D-modules. When G=GLn, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra (DAHA) SHq,t.
Original languageEnglish
Pages (from-to)4711-4748
Number of pages38
JournalSelecta Mathematica (New Series)
Volume24
Issue number5
Early online date26 Jul 2018
DOIs
Publication statusPublished - Nov 2018

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