Cardy limit of the 3d superconformal index

We study the superconformal index $Z(q)$ of 3d $\mathcal{N}=2$ gauge theories in Cardy-like limits $\beta = \log \tfrac{1}{q} \to 0^+$, 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 $multiscale\ decomposition$ 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 $Lorentzian\ factorization$ formula that establishes complete screening. A key insight is the use of $Poisson\ resummation$, 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.