The deconfinernent and Hagedorn phase transitions 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 Sd-1 x 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 Hagedom temperature or by two continuous phase transitions, the first of which occurs at the Hagedorn temperature. These phase transitions appear to be continuously connected to the usual flat space deconfinement transition in the case of confining gauge theories, and to the Hawking-Page nucleation of AdS(5) black holes in the case of the N = 4 supersymmetric Yang-Mills theory. Our analysis proceeds by first reducing the Yang-Mills partition function to a (0 + 0)-dimensional integral over a unitary matrix U, which is the holonomy (Wilson loop) of the gauge field around the thermal time circle in Euclidean space; deconfinement transitions are large N transitions in this matrix integral. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.