Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface

Abstract

This paper studies supersymmetric ground states of 3d $\mathcal{N}=4$ supersymmetric gauge theories on a Riemann surface of genus $g$. There are two distinct spaces of supersymmetric ground states arising from the $A$ and $B$ type twists on the Riemann surface, which lead to effective supersymmetric quantum mechanics with four supercharges and supermultiplets of type $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ respectively. We compute the space of supersymmetric ground states in each case, graded by flavour and R-symmetries and in different chambers for real mass and FI parameters, for a large class of supersymmetric gauge theories. The results are formulated geometrically in terms of the Higgs branch geometry. We perform extensive checks of compatibility with the twisted index and mirror symmetry.

50 pages, SciPost version

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• Batalin--Vilkovisky quantization and supersymmetric twists
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Truncated affine Rozansky--Witten models as extended defect TQFTs