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

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

doi:10.1007/jhep08(2019)120

Alejandro Cabo-Bizet, Dario Martelli + Show 2 more

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https://doi.org/10.1007/jhep08(2019)120

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Abstract

We show that the superconformal index of mathcal{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 AdS5 theory.

Highlights

This observation suggested the existence of a thermodynamic principle which leads to the above extremization formula
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
The other method is to analyze the saddle points at large N and 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 AdS5 theory

Summary

The Cardy-like limit of the superconformal index

The Cardy-like (“high-temperature”) limit of the superconformal index was studied in [21, 27] using effective field theory arguments on S3 × S1 backgrounds, and in [22] using asymptotic properties of the elliptic gamma functions. Assuming real values of the fugacities and taking the limit of small chemical potentials σ, τ , these works found a universal exponential contribution to the index weighted by the ’t Hooft anomaly TrR ∝ (a − c) (this can receive corrections in special instances [22]). Assuming real values of the fugacities and taking the limit of small chemical potentials σ, τ , these works found a universal exponential contribution to the index weighted by the ’t Hooft anomaly TrR ∝ (a − c) (this can receive corrections in special instances [22]). It follows that in this limit the asymptotic growth of the index at large N is not enough to reproduce the Bekenstein-Hawking entropy of holographically dual AdS5 black holes.

Derivation of the saddle point

The large-N saddle points of the superconformal index

The continuum matrix model and its saddle points at large N

Large-N saddle points of the discrete model

Saddle-point evaluation of the integral and the Cardy-like limit

A Saddle points with partial or no clumping for generic quivers