Superconformal indices at large $N$ and the entropy of AdS$_5$ $\times$ SE$_5$ black holes

TL;DR

The paper develops a unified large-$N$ framework for counting 1/16 BPS states in 4d $ ext{N}=1$ holographic theories via the Bethe Ansatz formulation of the superconformal index, extending beyond equal angular momenta to arbitrary charges. It identifies a universal BA contribution that dominates at large $N$, whose Legendre transform reproduces the entropy of BPS Kerr–Newman black holes in AdS$_5 imes S^5$ and, more generally, in AdS$_5 imes { m SE}_5$ via entropy functions built from 't Hooft anomaly data for toric quivers. The authors provide explicit checks: for $ ext{N}=4$ SYM and for toric quivers, including the conifold, they derive entropy functions that match the BPS near-horizon geometries; they also construct and analyze the near-horizon geometry of supersymmetric black holes in AdS$_5 imes T^{1,1}$ using 5d–4d reductions and the attractor mechanism, achieving full agreement with field theory predictions. Overall, the work offers a microscopic, large-$N$ derivation of BH microstates across a broad class of AdS$_5$ backgrounds and highlights a universal structure governing the entropy from the superconformal index.

Abstract

The large $N$ limit of the four-dimensional superconformal index was computed and successfully compared to the entropy of a class of AdS$_5$ black holes only in the particular case of equal angular momenta. Using the Bethe Ansatz formulation, we compute the index at large $N$ with arbitrary chemical potentials for all charges and angular momenta, for general $\mathcal{N}=1$ four-dimensional conformal theories with a holographic dual. We conjecture and bring some evidence that a particular universal contribution to the sum over Bethe vacua dominates the index at large $N$. For $\mathcal{N}=4$ SYM, this contribution correctly leads to the entropy of BPS Kerr-Newman black holes in AdS$_5 \times S^5$ for arbitrary values of the conserved charges, thus completing the microscopic derivation of their microstates. We also consider theories dual to AdS$_5 \times \mathrm{SE}_5$, where SE$_5$ is a Sasaki-Einstein manifold. We first check our results against the so-called universal black hole. We then explicitly construct the near-horizon geometry of BPS Kerr-Newman black holes in AdS$_5 \times T^{1,1}$, charged under the baryonic symmetry of the conifold theory and with equal angular momenta. We compute the entropy of these black holes using the attractor mechanism and find complete agreement with the field theory predictions.

Figures (1)

• Figure 1: Fundamental strips for $[\Delta]_\omega$ and $[\Delta]'_\omega$. The function $[\Delta]_\omega$ is the restriction of $\Delta$ mod $1$ to the region $\mathop{\mathrm{\mathbb{I}m}}\nolimits(-1/\omega) > \mathop{\mathrm{\mathbb{I}m}}\nolimits(\Delta/\omega)>0$ (in yellow, on the left), while $[\Delta]'_\omega$ is the restriction of $\Delta$ mod $1$ to the region $0 > \mathop{\mathrm{\mathbb{I}m}}\nolimits(\Delta/\omega) > \mathop{\mathrm{\mathbb{I}m}}\nolimits(1/\omega)$ (in blue, on the right).