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Selecta Mathematica

, Volume 24, Issue 5, pp 4711–4748 | Cite as

Quantum character varieties and braided module categories

  • David Ben-Zvi
  • Adrien Brochier
  • David Jordan
Open Access
Article

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 \({\mathbb {SH}}_{q,t}\).

Mathematics Subject Classification

17B37 16T99 

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TexasAustinUSA
  2. 2.MPIMBonnGermany
  3. 3.School of MathematicsUniversity of EdinburghEdinburghUK

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