Thermal N=4 supersymmetric Yang-Mills theory as a 2D Coulomb gas

We consider N=4 supersymmetric Yang-Mills theory with SU(N) gauge group at large N and at finite temperature on a spatial S3. We show that, at finite weak ’t Hooft coupling, a sector of the theory is naturally described as a two-dimensional Coulomb gas of complex eigenvalues, ΦJ+iA0, valued on the cylinder. In the low temperature confined phase the eigenvalues condense onto a strip encircling the cylinder, while the high temperature deconfined phase is characterized by an ellipsoidal droplet of eigenvalues