Cardy limit of the 3d superconformal index

Abstract

We study the superconformal index $Z(q)$ of 3d $\mathcal{N}=2$ gauge theories in Cardy-like limits $\beta =\log \frac{1}{q}\to 0^{+}$, extending techniques recently developed in the 4d $\mathcal{N}=1$ context. For theories with vectorlike matter content we find on the first sheet ($q\to 1$) that $Z(q)\sim {\beta }^{-\#}$, where the exponent $\#$ is determined by a $multiscaledecomposition$ of the BPS moduli space appearing in the localization formula for the index. On the second sheet ($q\to e^{2\pi i})$ we find $Z(q)\sim e^{\#/\beta }$, and that the long-standing puzzle of apparent gauge-enhancing saddles is resolved (in the absence of Chern--Simons couplings) via a novel $Lorentzianfactorization$ formula that establishes complete screening. A key insight is the use of $Poissonresummation$, which streamlines the asymptotic analysis, sharpens the link to Kaluza--Klein effective field theory, and provides a dual description of parts of the BPS moduli space in terms of punctured surfaces. The Lorentzian factorization formula also emerges from Poisson resummation, though applied after a contour crossing in moduli space. This, in turn, hints at a correspondence between 3d monopoles and vortices via 2d duality.

5 figures, a 12-page summary in the introduction section. v2: minor corrections