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Quantum character varieties and braided module categories

Selecta Mathematica volume 24, pages 4711–4748 (2018)Cite this 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 \({\mathbb {SH}}_{q,t}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Texas, Austin, TX, 78712-0257, USA

    David Ben-Zvi

  2. MPIM, Bonn, Germany

    Adrien Brochier

  3. School of Mathematics, University of Edinburgh, Edinburgh, UK

    David Jordan

Authors
  1. David Ben-Zvi
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  2. Adrien Brochier
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  3. David Jordan
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Correspondence to David Jordan.

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Ben-Zvi, D., Brochier, A. & Jordan, D. Quantum character varieties and braided module categories. Sel. Math. New Ser. 24, 4711–4748 (2018). https://doi.org/10.1007/s00029-018-0426-y

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Mathematics Subject Classification

  • 17B37
  • 16T99