High-temperature expansion of supersymmetric partition functions

Di Pietro and Komargodski have recently demonstrated a four-dimensional counterpart of Cardy's formula, which gives the leading high-temperature (beta -> 0) behavior of supersymmetric partition functions Z(SUSY) (beta). Focusing on superconformal theories, we elaborate on the subleading contributions to their formula when applied to free chiral and U(1) vector multiplets. In particular, we see that the high-temperature expansion of ln Z(SUSY) (beta) terminates at order beta(0). We also demonstrate how their formula must be modified when applied to SU(N) toric quiver gauge theories in the planar (N -> infinity) limit. Our method for regularizing the one-loop determinants of chiral and vector multiplets helps to clarify the relation between the 4d N = 1 superconformal index and its corresponding supersymmetric partition function obtained by path-integration.