High-Temperature Expansion of Supersymmetric Partition Functions

TL;DR

This paper explores the high-temperature behavior of supersymmetric partition functions in four-dimensional superconformal theories, revealing the precise subleading contributions and modifications for large N gauge theories, thereby clarifying their relation to superconformal indices.

Contribution

It provides a detailed analysis of the subleading terms in the high-temperature expansion of supersymmetric partition functions and introduces a regularization method for one-loop determinants.

Findings

High-temperature expansion terminates at order β^0 for free multiplets.

Modifications are necessary for SU(N) quiver gauge theories in the large N limit.

Clarifies the relation between superconformal index and partition function.

Abstract

Di Pietro and Komargodski have recently demonstrated a four-dimensional counterpart of Cardy's formula, which gives the leading high-temperature ($\beta\rightarrow{0}$) 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 ($N\rightarrow\infty$) 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.