Large-N CPN -1 sigma model on a Euclidean torus: uniqueness and stability of the vacuum

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, beta), beta being the inverse temperature. We find that the system has a unique homogeneous phase, with the CPN -1 fields n(i) acquiring a dynamically generated mass (lambda) >= Lambda (2) (analogous to the mass gap of SU(N ) Yang-Mills theory in 4D), for any beta 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 R-2, 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 theta dependence and different limits involving N , beta and L are briefly discussed.