Equivariant Verlinde formula from fivebranes and vortices

TL;DR

By embedding complex Chern-Simons theory with G_C on Seifert manifolds into string theory, the paper introduces a β-deformation that regularizes the theory and defines an equivariant Verlinde formula as a graded dimension over the Hitchin moduli space. It presents five independent frameworks to compute this graded dimension: a β-deformed 3d Lens space theory, 3d-3d/Hitchin integral, a β-deformed 2d equivariant G/G gauged WZW theory, a β-deformed 2d gauged WZW-matter theory, and a Bethe/gauge perspective, with cross-checks via localization and dualities. It then extends to a family of 2d TQFTs labeled by R, clarifying the R=2 (equivariant G/G) and R=0 (GWZWM) cases and their connections to vortex moduli and Grassmannian quantum cohomology. Finally, it develops a t-deformation and categorification of the Verlinde algebra, including an equivariant Higgs vertex and Bethe-state structures, linking fusion rules to Hitchin data and Bethe equations.

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

Figures (7)

• Figure 1: A genus-2 Riemann surface can be decomposed into two pairs of pants.
• Figure 2: ${\mathbb C \mathbb P}^1$ can be viewed as the total space of a circle fibration over an interval, with degenerate fibers at the endpoints of the interval.
• Figure 3: The Lens space $L(k,1)$ can be viewed as the total space of a 2-torus fibered over an interval. Near each endpoint of the interval, a particular cycle of the torus degenerates.
• Figure 4: The NS5-D3-(1,$k$) brane system in type IIB string theory.
• Figure 5: The (1,$k$)-brane in figure \ref{['fig:NS5-D3-(1,k)']} is a bound state of an NS5-brane and $k$ D5-branes.