AdS${}_7$ Black Holes and Holography

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.