High-Temperature Expansion of Supersymmetric Partition Functions

Abstract

Di Pietro and Komargodski have recently demonstrated a four-dimensional counterpart of Cardy's formula, which gives the leading high-temperature ($\beta \to 0$) 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 ($N\to \infty$) limit. Our method for regularizing the one-loop determinants of chiral and vector multiplets helps to clarify the relation between the 4d $\mathcal{N}=1$ superconformal index and its corresponding supersymmetric partition function obtained by path-integration.

15 pages plus appendices; v2: minor modifications and a "Note added"; v3: presentation improved and minor errors in app B corrected

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