Universal Cardy-Like Behavior of 3D Partition Functions from Supersymmetric Localization

We investigate 3d $\mathscr{N}=2$ supersymmetric gauge theories on $S^1 \times S^2$ and the corresponding 2d effective field theories arising in the limit of small ratio of radii, $\beta=R_{S^1}/R_{S^2}\to 0$. We evaluate the exact partition function of these theories in the framework of supersymmetric localization on curved backgrounds. As a result, we establish a finite-$N$ map between a particular, superconformal-index-inspired partition function and the topologically twisted index. Taking the large-$N$ limit of the partition functions, we reproduce the entropy functions of either spherically symmetric, magnetically charged, or rotating, electrically charged asymptotically AdS$_4$ black holes. We then recast the problem of evaluating the 3d partition functions directly in the framework of rigid supersymmetry. By carefully tracking the background fields, we find that in the small-$\beta$ limit, the partition functions of these 3d large-$N$ superconformal field theories have a universal behavior related to the coefficients of the R-symmetry or flavor symmetry 2-point current correlation functions, thus obtaining a universal Cardy-like formula for 3d $\mathscr{N}=2$ superconformal field theories.