The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories

We demonstrate that weakly coupled, large N , d-dimensional SU (N ) gauge theories on a class of compact spatial manifolds (including S d-1 time) undergo deconfinement phase transitions at temperatures proportional to the inverse length scale of the manifold in question. The low temperature phase has a free energy of order one, and is characterized by a stringy (Hagedorn) growth in its density of states. The high temperature phase has a free energy of order N 2 . These phases are separated either by a single first order transition that generically occurs below the Hagedorn temperature or