Supersymmetric AdS5 Black Holes and the Superconformal Index

We review recent developments on the superconformal index of N=4 SU(N) supersymmetric Yang-Mills theory. Thanks to the AdS/CFT correspondence, this can be used to address the problem of microstate counting for 1/16-BPS black holes in AdS5. The index may be thought of as a Euclidean partition function of the theory on S^3XS^1 with complex chemical potentials which we map via AdS/CFT to a sum over Euclidean gravity solutions. In the large-N limit, the index can be studied using a matrix model with complex couplings. We exhibit a phenomenon in complex matrix models which we call "delayed deconfinement". It turns out that when the action is complex, due to destructive interference, tachyonic modes do not necessarily condense.  We determine the deconfinement line of N=4 SYM and find remarkable agreement with the prediction of gravity. Finally, we study the global properties of the Euclidean partition function on S^3XS^1 with complex fugacities. This admits a novel Cardy-like limit analogous to a high-temperature limit in CFT. We present a formula for the asymptotic behavior of the index in this limit which can be derived uniquely in terms of the contribution from superconformal anomalies.