Equivariant Verlinde formula from fivebranes and vortices

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 Σ×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 Σ×S^1 and (4) the equivariant index of a spin^c Dirac operator on the moduli space of Higgs bundles.