Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space

Abstract

We study the twisted indices of $\mathcal{N}=4$ supersymmetric gauge theories in three dimensions on spatial $S^2$ with an angular momentum refinement. We demonstrate factorisation of the index into holomorphic blocks for the $T[SU(N)]$ theory in the presence of generic fluxes and fugacities. We also investigate the relation between the twisted index, Hilbert series and the moduli space of vortices. In particular, we show that each holomorphic block coincides with a generating function for the ${\chi }_t$ genera of the moduli spaces of "local" vortices. The twisted index itself coincides with a corresponding generating function for the ${\chi }_t$ genera of moduli spaces of "global" vortices in agreement with a proposal of Bullimore et. al. We generalise this geometric interpretation of the twisted index to include fluxes and Chern-Simons levels. For the $T[SU(N)]$ theory, the relevant moduli spaces are the local and global versions of Laumon space respectively and we demonstrate the proposed agreements explicitly using results from the mathematical literature. Finally, we exhibit a precise relation between the Coulomb branch Hilbert series and the Poincaré polynomials of the corresponding vortex moduli spaces.

arXiv comment56 pages, 4 figures. v2: published version, references added

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Cited by in corpus (8)

• Boundaries, Vermas, and Factorisation
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• Blocks and Vortices in the 3d ADHM Quiver Gauge Theory
• Topologically twisted index of $T[SU(N)]$ at large $N$
• Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function
• Vortex counting and the quantum Hall effect
• The colored Jones polynomials as vortex partition functions
• Non-Perturbative Schwinger-Dyson Equations for 3d $\mathcal{N}=4$ Gauge Theories