CPN sigma model on a finite interval revisited

In this paper we will revisit the large N solution of the CPN sigma model on a finite interval of length L. We will find a family of boundary conditions for which the large N saddle point can be found analytically. For a certain choice of the boundary conditions the theory has only one phase for all values of L. Also, we will provide an example when there are two phases: for large L there is a standard phase with an unbroken U(1) gauge symmetry and for small L there is a Higgs phase with a broken gauge symmetry.