Quantum character varieties and braided module categories
By David Ben-Zvi, Adrien Brochier, D. Jordan. Selecta Mathematica, 2016
90 citations recorded by the source corpus.
We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants ∫SA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _S{\mathcal {A}}$$\end{document} of a surface S, determined by the choice of a braided tensor category A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}, 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 A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}, 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 A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}-modules are objects of the torus category ∫T2A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{T^2}{\mathcal {A}}$$\end{document}. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of A=RepqG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}={\text {Rep}}_{q}G$$\end{document} with the category Dq(G/G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}_q(G/G)$$\end{document}-mod of equivariant quantum D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-modules. When G=GLn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=GL_n$$\end{document}, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra SHq,t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {SH}}_{q,t}$$\end{document}.
Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics
View Paper: https://www.semanticscholar.org/paper/d732e575ce2d264c2fb93be70c60061f7d2935e4
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Citations in the corpus
- Braided Picard groups and graded extensions of braided tensor categories (2020, background): “ym(B) symmetric pseudo-monoid Table 1: Extensions B ⊂ C as pseudo-monoids in a monoidal 2-category M. denote Modsym(E) the symmetric monoidal 2-categoryof symmetric E-module categories1. By definition [5, 2], a braided B-module category is equipped with a natural collection of isomorphisms coherently”
- The finiteness conjecture for skein modules (2019, methodology): “The functor ZA(−) was studied extensively in [8,9].”
- Betti Geometric Langlands (2016, background): “algebra Aq(S) recovers the quantum function algebra Oq(G∨) (with its conjugation action) in the case of an annulus, and the algebra Dq(G∨) of quantum differential operators for the punctured torus. In [BBJ2] the category QC q(G∨/G∨) = R S1 Rep G ∨ is endowed with two explicit tensor structures, a bra”