The superconformal index of N=4 super Yang-Mills theory on a three-sphere is captured by a unitary matrix model with purely double trace operators in the action. The AdS/CFT correspondence predicts that this index should have exponential growth at large charges and large N, corresponding to the 1/16-BPS black hole (BH) in AdS5. I will show how the matrix model gives rise to this expected BH growth as well as an infinite number of new phases. In particular, I will introduce a deformation of the matrix model which allows us to solve it at large N. The deformation has interesting relations with the Bloch-Wigner dilogarithm, a function introduced by number theorists.
I will then show how this matrix model can be expressed in terms of a system of free fermions in a certain ensemble. Integrating out the fermions and averaging over the ensemble leads to a convergent expansion as a series of determinants, showing how giant gravitons in the dual AdS5 are encoded in the gauge theory.
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