Equivariant Verlinde formula from fivebranes and vortices

We study complex Chern-Simons theory on a Seifert manifold M3 by embedding it into string theory. We show that complex Chern-Simons theory on M3 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 × S1 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 vortex moduli spaces, 3) complex Chern-Simons theory on Σ × S1 and 4) the equivariant index of a spinc Dirac operator on the moduli space of Higgs bundles. CALT-TH-2014-171 ar X iv :1 50 1. 01 31 0v 1 [ he pth ] 6 J an 2 01 5