The superconformal index and black hole instabilities

The superconformal index of \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory with gauge group U(N) has provided powerful insights into the entropy of supersymmetric black holes in AdS₅ × S⁵, including some sub-leading logarithmic and non-perturbative corrections. Recently, the phase space of supersymmetric solutions has been argued to contain configurations other than the asymptotically AdS₅ black hole. Such configurations include the so-called grey galaxies where the black hole at the center is surrounded by a gas of gravitons. By numerically evaluating the superconformal index of \( \mathcal{N} \) = 4 supersymmetric Yang-Mills at small values of N, we detect systematic deviations from the entropy of black holes with two distinct angular momenta. We find that the giant graviton expansion of the index is a numerically efficient way of evaluating the index that complements the direct character evaluation and allows for explicit access to N ≤ 15 with up to two giant gravitons in the expansion. We find it remarkable that a supersymmetric quantity in field theory, usually thought of as a rigid counting observable, indeed contains information about different phases in the space of supersymmetric solutions on the gravity side.