Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps

Abstract

We explore the geometric interpretation of the twisted index of 3d $\mathcal{N}=4$ gauge theories on $S^1\times \Sigma$ where $\Sigma$ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d $\mathcal{N}=4$ mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters.

arXiv comment65 pages, 2 figures, minor corrections, version to appear in JHEP

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Cited by in corpus (18)

• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• Mirror symmetry and line operators
• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs
• Boundaries, Vermas, and Factorisation
• Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
• Quantum K theory of symplectic Grassmannians
• Blocks and Vortices in the 3d ADHM Quiver Gauge Theory
• Quantum K-Theory of Calabi-Yau Manifolds
• 3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties
• Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space
• Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs
• Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface
• Vortex counting and the quantum Hall effect
• Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg Algebras
• ${\text{AdS}}_4$ Holography and the Hilbert Scheme
• The colored Jones polynomials as vortex partition functions
• Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials