Abstract
We analyze a set of contributions to the superconformal index of 4d \U0001d4a9 = 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(N0) terms, including those exponentially suppressed in the Cardy limit, and show that there are no 1/N corrections beyond them. Under certain assumptions, it implies that the gravitational perturbative series around these black hole saddles is 1-loop exact.
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