Large-N \u2102\u2119N \u22121 sigma model on a Euclidean torus: uniqueness and stability of the vacuum

Abstract

In this paper we examine analytically the large-N gap equation and its solution for the 2D ℂℙN −1 sigma model defined on a Euclidean spacetime torus of arbitrary shape and size (L, β), β being the inverse temperature. We find that the system has a unique homogeneous phase, with the ℂℙN −1 fields ni acquiring a dynamically generated mass (λ) ≥ Λ2 (analogous to the mass gap of SU(N ) Yang-Mills theory in 4D), for any β and L. Several related topics in the recent literature are discussed. One concerns the possibility, which turns out to be excluded according to our analysis, of a “Higgs-like” — or deconfinement — phase at small L and at zero temperature. Another topics involves “soliton-like” (inhomogeneous) solutions of the generalized gap equation, which we do not find. A related question concerns a possible instability of the standard ℂℙN −1 vacuum on R2, which is shown not to occur. In all cases, the difference in the conclusions can be traced to the existence of certain zeromodes and their proper treatment. The ℂℙN −1 model with twisted boundary conditions is also analyzed. The θ dependence and different limits involving N , β and L are briefly discussed.