The large-$N$ limit of the 4d $\mathcal{N}=1$ superconformal index

TL;DR

This paper analyzes the large-$N$ limit of the superconformal index in 4d $ =1$ theories with quiver descriptions, revealing universal saddle points related to black hole solutions and anomaly coefficients.

Contribution

It introduces a systematic method to evaluate the superconformal index at large $N$ using elliptic extensions and identifies universal saddle points linked to anomalies and holographic black holes.

Findings

Universal saddle points controlled by anomaly coefficients.

Connection between saddle points and supersymmetric black holes.

Universal cubic form of the effective action with flavor chemical potentials.

Abstract

We systematically analyze the large-$N$ limit of the superconformal index of $\mathcal{N}=1$ superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate using the recently-developed technique of elliptic extension. This technique allows us to easily evaluate the integral as a sum over saddle points of an effective action in the limit where the rank of the gauge group is infinite. For a generic quiver theory under consideration, we find a special family of saddles whose effective action takes a universal form controlled by the anomaly coefficients of the theory. This family includes the known supersymmetric black hole solution in the holographically dual AdS$_5$ theories. We then analyze the index refined by turning on flavor chemical potentials. We show that, for a certain range of chemical potentials, the effective action again takes a universal cubic form that is controlled by the anomaly coefficients of the theory. Finally, we present a large class of solutions to the saddle-point equations which are labelled by group homomorphisms of finite abelian groups of order $N$ into the torus.