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The superconformal index of the $\mathcal{N}=4$ SU(N) supersymmetricYang-Mills theory counts the 1/16-BPS states in this theory, and has been usedvia the AdS/CFT correspondence to count black hole microstates of 1/16-BPSblack holes. On one hand, this index may be related to the Euclidean partitionfunction of the theory on $S^3\times S^1$ with complex chemical potentials,which maps by the AdS/CFT correspondence to a sum over Euclidean gravitysolutions. On the other hand, the index may be expressed as a sum oversolutions to Bethe Ansatz Equations (BAEs). We show that the known solutions tothe BAEs that have a good large $N$ limit, for the case of equal chemicalpotentials for the two angular momenta, have a one-to-one mapping to (complex)Euclidean black hole solutions on the gravity side. This mapping captures boththe leading contribution from the classical gravity action (of order $N^2$), aswell as non-perturbative corrections in $1/N$, which on the gravity side arerelated to wrapped D3-branes. Some of the BA solutions map to orbifolds of thestandard Euclidean black hole solutions (that obey exactly the same boundaryconditions as the other solutions). A priori there are many more gravitationalsolutions than Bethe Ansatz solutions, but we show that by considering thenon-perturbative effects, the extra solutions are ruled out, leading to aprecise match between the solutions on both sides.

A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index