A second order deconfinement transition for largeN2+1 dimensional Yang-Mills theory on a smallS²

Abstract

We study the thermodynamics of large N pure 2+1 dimensional Yang-Mills theory on a small spatial S 2 . By studying the effective action for the Polyakov loop order parameter, we show analytically that the theory has a second order deconfinement transition to a phase where the eigenvalue distribution of the Polyakov loop is non-uniform but still spread over the whole unit circle. At a higher temperature, the eigenvalue distribution develops a gap, via an additional third-order phase transition. We discuss possible forms of the full phase diagram as a function of temperature and sphere radius. Our results together with extrapolation of lattice results relevant to the large volume limit imply the existence of a critical radius in the phase diagram at which the deconfinement transition switches from second order to first order. We show that the point at the critical radius and temperature can be either a tricritical point with universal behavior or a triple point separating three distinct phases.

Phases of large N vector Chern-Simons theories on S 2 × S 1

“…We have found the exact solution to the saddle point 8 In the case that the base manifold is an S 2 , as studied in our paper, the exclusion of such configurations follows from the fact that the measure factor in (1.1) eliminates their contributions. The generalization of (1.6) to the partition function of Chern-Simons theory on Σg × S 1 where Σg is a genus g manifold of arbitrary metric is given by the formula (4.6).…”

“…(1.1) may be thought of as a Landau Ginzburg or Wilsonian description of the holonomy U , the lightest degree of freedom of the finite temperature field theory. The effective potential V YM (U ) was computed in free gauge theories [5,2]; it has also been evaluated at higher orders in perturbation theory in special examples [6,7,8,9]. In all these examples V YM (U ) is an attractive potential for the eigenvalues of the unitary matrix.…”

“…As the summation in (1.6) effectively excludes terms with coincident n m 's, 8 it follows that the maximum number of eigenvalues in an interval ∆α in the summation in (1.6) is given by…”

Duality and higher temperature phases of large N Chern-Simons matter theories on S 2 × S 1

“…The effective potential VY M (U ) was computed in free gauge theories[34,35]; it has also been evaluated at higher orders in perturbation theory in special examples[36,37,38,39]. At least in perturbation theory[35] and perhaps beyond[40,41], the potential VY M (U ) is an analytic function of U .…”

Phases of a two-dimensional large-Ngauge theory on a torus

“…Because of the difficulty of the analysis of Yang-Mills theory, only weakly coupled Yang-Mills theories on S 2 and S 3 [64,65] have been studied. In these cases, all the spatial components of the gauge field have a mass proportional to 1/R, where R is the radius of the sphere.…”