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

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.