High-temperature asymptotics of the 4d superconformal index

TL;DR

This work analyzes the high-temperature asymptotics of the 4d superconformal index for unitary non-chiral Lagrangian SCFTs, recasting the index as an elliptic hypergeometric integral whose beta→0 limit localizes via a Rains-function-driven effective potential. It derives a general framework where the leading exponential is governed by the Di Pietro–Komargodski-type term involving $c-a$ and a subleading piece from the minimum of the Rains function, with a logarithmic factor tied to the dimension of the quantum Coulomb branch. The thesis applies this to numerous finite-N theories and large-N toric quivers, obtaining precise asymptotics, and linking them to holographic Weyl anomalies, $S^3$-partition function divergences, and Coulomb-branch dynamics on $R^3 imes S^1$. It provides both concrete analytical results for exemplars (e.g., A_k, SO(2N+1), ISS, BCI, class-S theories) and general theorems governing the large-N limit via multi-trace vs single-trace indices, with implications for dualities, AdS/CFT checks, and the microscopic accounting of holographic states. The work thus connects intricate special-function techniques with physical observables in supersymmetric gauge theories and their gravity duals, offering a comprehensive toolkit for probing non-perturbative dynamics through high-temperature index asymptotics.

Abstract

The superconformal index of a typical Lagrangian 4d SCFT is given by a special function known as an elliptic hypergeometric integral (EHI). The high-temperature limit of the index corresponds to the hyperbolic limit of the EHI. The hyperbolic limit of certain special EHIs has been analyzed by Eric Rains around 2006; extending Rains's techniques, we discover a surprisingly rich structure in the high-temperature limit of a (rather large) class of EHIs that arise as the superconformal index of unitary Lagrangian 4d SCFTs with non-chiral matter content. Our result has implications for $\mathcal{N}=1$ dualities, the AdS/CFT correspondence, and supersymmetric gauge dynamics on $R^3\times S^1$. We also investigate the high-temperature asymptotics of the large-N limit of the superconformal index of a class of holographic 4d SCFTs (described by toric quiver gauge theories with SU(N) nodes). We show that from this study a rather general solution to the problem of holographic Weyl anomaly in AdS$_5$/CFT$_4$ at the subleading order (in the 1/N expansion) emerges. Most of this dissertation is based on published works by Jim Liu, Phil Szepietowski, and the author. We include here a few previously unpublished results as well, one of which is the high-temperature asymptotics of the superconformal index of puncture-less SU(2) class-$\mathcal{S}$ theories.

Figures (7)

• Figure 1: The Rains function of the $A_1$ SU($3$) theory---also known as SU($3$) SQCD.
• Figure 2: The Rains function of the SO($5$) SQCD.
• Figure 3: The Rains function of the SU($2$)$\times$SU($2$) orbifold theory.
• Figure 4: The Rains function of the SU($2$) ISS theory.
• Figure 5: The Rains function of the SO($5$) BCI theory.