| Summary: | Abstract We construct large N saddle points of the matrix model for the N $$ \mathcal{N} $$ = 4 Yang- Mills index dual to the BPS black holes in AdS 5 × S 5, in two different setups. When the two complex chemical potentials for the angular momenta are collinear, we find linear eigenvalue distributions which solve the large N saddle point equation. When the chemical potentials are not collinear, we find novel solutions given by areal eigenvalue distributions after slightly reformulating the saddle point problem. We also construct a class of multi-cut saddle points, showing that they sometimes admit nontrivial filling fractions. As a byproduct, we find that the Bethe ansatz equation emerges from our saddle point equation.
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