High-Temperature Expansion of Supersymmetric Partition Functions

TL;DR

This work refines the high-temperature analysis of four-dimensional SUSY partition functions by extending Cardy-like ideas to 4d SCFTs. It shows that for $Z^{SUSY}(\beta)$ the leading $\frac{16\pi^2(c-a)}{3\beta}$ term is supplemented by a universal $-4(2a-c)\ln(\beta/2\pi)$ and a $Z_{3d}$-dependent constant, with all perturbative corrections truncating at $O(1)$ and non-perturbative $e^{-1/\beta}$ pieces. The authors derive explicit subleading expansions for free chiral and U(1) vector multiplets, relate the index to the SUSY partition function, and then extend the analysis to large-$N$ toric quivers, where the planar limit modifies the leading and subleading terms in a geometrically and combinatorially rich way. The results illuminate how central charges, dual geometry, and adjoint matter enter the high-temperature behavior, and they clarify how to extract universal data from indices, with potential implications for holography and beyond. Together, these findings bolster the understanding of universal vs. non-universal aspects of SUSY partition functions at high temperature and pave the way for broader applications to non-Lagrangian and large-$N theories.

Abstract

Di Pietro and Komargodski have recently demonstrated a four-dimensional counterpart of Cardy's formula, which gives the leading high-temperature ($β\rightarrow{0}$) behavior of supersymmetric partition functions $Z^{SUSY}(β)$. 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 $\ln Z^{SUSY}(β)$ terminates at order $β^0$. 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.