Comments on twisted indices in 3d supersymmetric gauge theories

Abstract

We study three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on ${\Sigma }_g\times S^1$ with a topological twist along ${\Sigma }_g$, a genus-$g$ Riemann surface. The twisted supersymmetric index at genus $g$ and the correlation functions of half-BPS loop operators on $S^1$ can be computed exactly by supersymmetric localization. For $g=1$, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe equations for the theory on ${\mathbb{R}}^2\times S^1$. This also provides a powerful and simple tool to study 3d $\mathcal{N}=2$ Seiberg dualities. Finally, we study A- and B-twisted indices for $\mathcal{N}=4$ supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the $S^2\times S^1$ twisted indices and the Hilbert series of $\mathcal{N}=4$ moduli spaces.

66 pages plus appendix; v2: corrected typos and added references

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