Quantum character varieties and braided module categories

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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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