R-charged AdS(5) black holes and large N unitary matrix models

Using the anti-de Sitter (AdS)/conformal field theories correspondence, we establish a correspondence between the intricate thermal phases of $R$-charged ${\mathrm{AdS}}_{5}$ black holes and the $R$-charge sector of the $N=4$ gauge theory, in the large $N$ limit. Integrating out all fields in the gauge theory, except the thermal Polyakov line, leads to an effective unitary matrix model. In the canonical ensemble, a logarithmic term is generated in the nonzero charge sector of the matrix model. This term is important for discussing various supergravity properties like (i) the nonexistence of a thermal AdS phase as a solution, (ii) the existence of a point of cusp catastrophe in the phase diagram, and (iii) the matching of saddle points and the critical exponents of supergravity and those of the effective matrix model. Our study is restricted to a unitary matrix model where the action depends only on $\mathrm{Tr}U\text{ }\mathrm{Tr}{U}^{\ifmmode\dagger\else\textdagger\fi{}}$.