Equivariant Verlinde formula from fivebranes and vortices

TL;DR

This paper connects complex Chern-Simons theory, vortex dynamics, and equivariant algebraic structures through string theory embeddings, revealing new relations and regularizations in topological quantum field theories.

Contribution

It introduces the equivariant Verlinde algebra and links it to quantum K-theory and complex Chern-Simons theory, expanding the mathematical framework of gauge theories.

Findings

Complex Chern-Simons theory is equivalent to a topologically twisted supersymmetric theory.

Dimensional reduction reveals vortex dynamics in 4D gauge theory.

New relations between equivariant Verlinde algebra, quantum K-theory, and complex Chern-Simons theory.

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 spin$^c$ 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 spin$^c$ Dirac operator on the moduli space of Higgs bundles.