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The asymptotic growth of states of the 4d N=1 superconformal index

Alejandro Cabo-Bizet, Davide Cassani, Dario Martelli, Sameer Murthy

TL;DR

This work demonstrates that the four-dimensional ${\cal N}=1$ superconformal index exhibits exponential growth in charges in a Cardy-like limit, matching the Bekenstein-Hawking entropy of supersymmetric AdS$_5$ black holes. It achieves this via two complementary saddle-point analyses: a finite-$N$ Cardy-like limit and a large-$N$ saddle-point approach, both of which identify a universal saddle at vanishing gauge holonomies controlled by anomaly data ${\rm Tr}R$ and ${\rm Tr}R^3$, and express the leading behavior in terms of central charges $a,c$. For ${\cal N}=4$ SYM and a broad class of ${\cal N}=1$ quivers, the leading contribution to $\log \mathcal{I}$ reproduces the BH entropy function, with subleading corrections captured by the Cardy-like expansion. These results support a modular-like structure in 4d SUSY partition functions and provide a universal entropy function for holographic SCFTs in the Cardy-like regime.

Abstract

We show that the superconformal index of N=1 superconformal field theories in four dimensions has an asymptotic growth of states which is exponential in the charges. Our analysis holds in a Cardy-like limit of large charges, for which the index is dominated by small values of chemical potentials. In this limit we find the saddle points of the integral that defines the superconformal index using two different methods. One method, valid for finite N, is to first take the Cardy-like limit and then find the saddle points. The other method is to analyze the saddle points at large N and then take the Cardy-like limit. The result of both analyses is that the asymptotic growth of states of the superconformal index exactly agrees with the Bekenstein-Hawking entropy of supersymmetric black holes in the dual AdS$_5$ theory.

The asymptotic growth of states of the 4d N=1 superconformal index

TL;DR

This work demonstrates that the four-dimensional superconformal index exhibits exponential growth in charges in a Cardy-like limit, matching the Bekenstein-Hawking entropy of supersymmetric AdS black holes. It achieves this via two complementary saddle-point analyses: a finite- Cardy-like limit and a large- saddle-point approach, both of which identify a universal saddle at vanishing gauge holonomies controlled by anomaly data and , and express the leading behavior in terms of central charges . For SYM and a broad class of quivers, the leading contribution to reproduces the BH entropy function, with subleading corrections captured by the Cardy-like expansion. These results support a modular-like structure in 4d SUSY partition functions and provide a universal entropy function for holographic SCFTs in the Cardy-like regime.

Abstract

We show that the superconformal index of N=1 superconformal field theories in four dimensions has an asymptotic growth of states which is exponential in the charges. Our analysis holds in a Cardy-like limit of large charges, for which the index is dominated by small values of chemical potentials. In this limit we find the saddle points of the integral that defines the superconformal index using two different methods. One method, valid for finite N, is to first take the Cardy-like limit and then find the saddle points. The other method is to analyze the saddle points at large N and then take the Cardy-like limit. The result of both analyses is that the asymptotic growth of states of the superconformal index exactly agrees with the Bekenstein-Hawking entropy of supersymmetric black holes in the dual AdS theory.

Paper Structure

This paper contains 13 sections, 124 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Cardy-like limit of the superconformal index
  3. Derivation of the saddle point
  4. $\mathcal{N}=4$ SYM with gauge group $G = SU(N)$
  5. Superconformal ${\cal N}=1$ quiver gauge theories
  6. The large-$N$ saddle points of the superconformal index
  7. The continuum matrix model and its saddle points at large $N$
  8. Large-$N$ saddle points of the discrete model
  9. Non-chiral quivers.
  10. Chiral quivers with equal R-charges.
  11. Other gauge groups.
  12. Saddle-point evaluation of the integral and the Cardy-like limit
  13. Saddle points with partial or no clumping for generic quivers

Figures (3)

  • Figure 1: The function $f$, determining the leading behavior of the effective potential.
  • Figure 2: The function $\tilde{f}(x)$ for a chiral field $Y$ in the $Y^{2,1}$ quiver gauge theory Benvenuti:2004dy, with R-charge $r=\frac{1}{3} \left(3 \sqrt{13}-9\right)\simeq 0.605$.
  • Figure 3: $N=8$$K$-gon solutions with $K=8,4,2,1$. The vertices represent clusters of $N/K$ eigenvalues in the complex plane ${\rm e}^{2\pi {\rm i} u}$.