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Equivariant Verlinde Formula from Fivebranes and Vortices

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Abstract

We study complex Chern–Simons theory on a Seifert manifold M 3 by embedding it into string theory. We show that complex Chern–Simons theory on M 3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern–Simons theory on \({\Sigma\times S^1}\) and (4) index of a spinc Dirac operator on the moduli space of flat connections to a new set of relations between (1) the “equivariant Verlinde algebra” for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern–Simons theory on \({\Sigma \times S^1}\) and (4) the equivariant index of a spinc Dirac operator on the moduli space of Higgs bundles.

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Authors and Affiliations

  1. Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, 91125, USA

    Sergei Gukov & Du Pei

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Sergei Gukov

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  1. Sergei Gukov
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Gukov, S., Pei, D. Equivariant Verlinde Formula from Fivebranes and Vortices. Commun. Math. Phys. 355, 1–50 (2017). https://doi.org/10.1007/s00220-017-2931-9

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