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 $\Sigma \times S^1$ 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.

56 pages, 7 figures; v2: misprints corrected, clarifications added, missing factors and terms restored in section 6.2

References in corpus (3)

• Ω-deformation and quantization
• Wilson loops in supersymmetric Chern-Simons-matter theories and duality
• Quantization via Mirror Symmetry

Cited by in corpus (15)

• Black hole microstates in AdS${}_4$ from supersymmetric localization
• Comments on twisted indices in 3d supersymmetric gauge theories
• Fivebranes and 3-manifold homology
• Supersymmetric partition functions on Riemann surfaces
• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• A 3d-3d appetizer
• Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality
• Refined 3d-3d Correspondence
• The equivariant Verlinde formula on the moduli of Higgs bundles
• Web of Seiberg-like dualities for 3d $\mathcal{N}=2$ quivers
• Mirror symmetry with branes by equivariant Verlinde formulae
• From Quantum Bäcklund Transforms to Topological Quantum Field Theory
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Chern-Simons Theory with Complex Gauge Group on Seifert Fibred 3-Manifolds
• Instanton-torus knot duality in 5d SQED and $SU(2)$ SQCD