Large-N ℂP N − 1 sigma model on a finite interval

We analyze the two-dimensional ℂPN − 1 sigma model defined on a finite space interval L, with various boundary conditions, in the large N limit. With the Dirichlet boundary condition at the both ends, we show that the system has a unique phase, which smoothly approaches in the large L limit the standard 2D ℂPN − 1 sigma model in confinement phase, with a constant mass generated for the ni fields. We study the full functional saddle-point equations for finite L, and solve them numerically. The latter reduces to the well-known gap equation in the large L limit. It is found that the solution satisfies actually both the Dirichlet and Neumann conditions.