Cardy Formula for 4d SUSY Theories and Localization

Abstract: We study 4d $\mathcal{N}=1$ supersymmetric theories on a compact Euclidean manifold of the form $S^1 \times\mathcal{M}_3$. Partition functions of gauge theories on this background can be computed using localization, and explicit formulas have been derived for different choices of the compact manifold $\mathcal{M}_3$. Taking the limit of shrinking $S^1$, we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of arXiv:1512.03376. The limit is given in terms of an effective potential for the holonomies around the $S^1$, whose minima determine the asymptotic behavior of the partition function. If the potential is minimized in the origin, where it vanishes, the partition function has a Cardy-like behavior fixed by $\mathrm{Tr}(R)$, while a nontrivial minimum gives a shift in the coefficient. In all the examples that we consider, the origin is a minimum if $\mathrm{Tr}(R) \leq 0$.