Free fermions and thermal AdS/CFT

The dynamics of finite temperature U(N) gauge theories on S³ can be described, at weak coupling, by an effective unitary matrix model. Here we present an exact solution to these models, for any value of N, in terms of a sum over representations. Taking the large N limit of this solution provides a new perspective on the deconfinement transition which is supposed to be dual to the Hawking-Page transition. The large N phase transition manifests itself here in a manner similar to the Douglas-Kazakov phase transition in 2d Yang-Mills theory. We carry out a complete analysis of the saddle representation in the simplest case involving only the order parameter TrU. We find that the saddle points corresponding to thermal AdS, the small black hole and the large black hole can all be described in terms of free fermions. They all admit a simple phase space description a la the BPS geometries of Lin, Lunin and Maldacena.