AdS$_7$ Black-Hole Entropy and 5D $\mathcal{N}=2$ Yang-Mills

We generalise the work of 1810.11442 for the case of AdS$_7$/CFT$_6$. Starting from the 2-equivalent charge, 3-equivalent rotation non-extremal black-hole solution in 7D gauged supergravity, we consider the supersymmetric and then the extremal limit and evaluate the associated thermodynamic quantities. Away from extremality, the black-hole solution becomes complex. The entropy is then given by the Legendre transform of the on-shell action with respect to two complex chemical potentials subject to a constraint. At the conformal boundary we derive the dual background and evaluate the corresponding partition function for the $A_{N-1}$ 6D (2,0) theory at large $N$ in a Cardy-like limit. This is carried out via a 5D $\mathcal N=2$ super Yang-Mills calculation on $S^5$. The gravitational on-shell action is found to be exactly reproduced by the boundary partition function at large $N$. We argue that this agreement puts strong constraints on the form of possible higher-derivative corrections to the 5D gauge theory that is used in the $S^5$ evaluation.