A note on the entropy of rotating BPS AdS$_7\times S^4$ black holes

TL;DR

The paper demonstrates that the entropy of BPS, rotating AdS7×S4 black holes can be obtained by extremizing a functional built from anomaly data of the 6D N=(2,0) theory, generalizing a similar AdS5 result. In the large-N limit, the relevant quantity is the equivariant integral $E^{(A_{N-1})}( abla_I, abla_i)$ of the six-dimensional anomaly polynomial, with $E^{(A_{N-1})}( abla_I, abla_i) = i \pi N^3 ((\nabla_1 \nabla_2)^2)/(12 \nabla_1 \nabla_2 \nabla_3)$. Entropy is obtained as the Legendre transform of $-E^{(A_{N-1})}$ minus charge couplings with the constraint $\sum_I \nabla_I + \sum_i \nabla_i = 1$, reproducing known black hole entropies for the two-charge single-rotation and three-rotation single-charge families. This links microscopic anomaly data to macroscopic black hole thermodynamics and suggests a Cardy-like mechanism in higher dimensions, with potential extensions to other AdS/CFT settings and holographic renormalization computations.

Abstract

In this note we show that the entropy of BPS, rotating, electrically charged AdS$_7 \times S^4$ black holes can be obtained by an extremization principle involving a particular combination of anomaly coefficients of the six-dimensional $\mathcal{N} = (2, 0)$ theory. This result extends our previous finding for BPS, rotating AdS$_5 \times S^5$ black holes.