Cardy formula for 4d SUSY theories and localization

We study 4d N=1 supersymmetric theories on a compact Euclidean manifold of the form S¹×M₃. 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 M₃. Taking the limit of shrinking S¹, we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of https://www.doi.org/10.1007/JHEP07(2016)025. The limit is given in terms of an effective potential for the holonomies around the S¹, 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 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 Tr(R)≤0.