The SUSY Index Beyond the Cardy Limit

We analyze a set of contributions to the superconformal index of 4d $\mathcal{N}=4$ $SU(N)$ super Yang-Mills using the Bethe Ansatz approach. These contributions dominate at the large $N$ limit, where their leading order in $N$ reproduces various supersymmetric Euclidean black hole saddles in the dual theory, and they also dominate for finite $N$ in high temperature Cardy-like limits. We compute the $O(N^0)$ terms, including those exponentially suppressed in the Cardy limit, and show that there are no $1\mathrm{/}N$ corrections beyond them. Under certain assumptions, it implies that the gravitational perturbative series around these black hole saddles is 1-loop exact.