AdS$_7$ Black Holes and Holography

TL;DR

This work analyzes the regularized on-shell action of asymptotically $AdS_7$ black holes with three angular momenta and two electric charges, showing that a holography-motivated choice of finite counterterms yields agreement with the large $N$ superconformal index of the dual 6d $\mathcal{N}=(2,0)$ $A_N$ SCFT on $S^1\times S^5$. It then extends the setup to a broad family of 7d black holes with horizons given by $L^{p,q,r}$ Sasaki–Einstein manifolds, whose on-shell actions likewise match localization results for the 6d SCFT on $S^1\times L^{p,q,r}$, and discusses a SUSY-limited, compact expression for the on-shell action. The paper also briefly generalizes AdS$_5$ backgrounds to 5d black holes with $L(p,q)$ lens-space horizons, highlighting holographic interpretations via lens-space indices. Overall, it strengthens the AdS$_7$/CFT$_6$ dictionary, clarifies the role of finite counterterms in holographic renormalization, and proposes concrete SUSY partition-function predictions for the 6d theory on novel boundary geometries.

Abstract

We present a detailed study of the regularized on-shell action of the recently found asymptotically AdS$_7$ black hole supergravity solution with three angular momenta and two electric charges. We show that a particular choice of finite counterterms in the holographic renormalization procedure yields a result for the on-shell action of the supersymmetric limit of the black hole solution which is in perfect agreement with the large $N$ limit of the superconformal index of the dual 6d $\mathcal{N}=(2,0)$ $A_{N}$ SCFT on $S^1\times S^5$. We also discuss a generalization of this 7d supergravity background to a new general family of black holes with horizons given by the $L^{p,q,r}$ 5d Sasaki-Einstein manifolds. The regularized on-shell action of these new 7d black holes is also in agreement with supersymmetric localization results for the 6d $\mathcal{N}=(2,0)$ SCFT on $S^1\times L^{p,q,r}$. Finally, we briefly discuss a generalization of known AdS$_5$ gauged supergravity backgrounds to new 5d black holes with $L(p,q)$ lens space horizon topologies.