Large-N ℂ ℙ N − 1 $$ \mathbb{C}{\mathrm{\mathbb{P}}}^{\mathrm{N}-1} $$ sigma model on a finite interval and the renormalized string energy

We continue the analysis started in a recent paper of the large-N two-dimensional $$ \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} $$ sigma model, defined on a finite space interval L with Dirichlet (or Neumann) boundary conditions. Here we focus our attention on the problem of the renormalized energy density $$ \mathrm{\mathcal{E}} $$ (x, Λ, L) which is found to be a sum of two terms, a constant term coming from the sum over modes, and a term proportional to the mass gap. The approach to $$ \mathrm{\mathcal{E}}\left(x,\varLambda,\ L\right)\to \frac{N}{4\pi }{\varLambda}^2 $$ at large LΛ is shown, both analytically and numerically, to be exponential: no power corrections are present and in particular no Luscher term appears. This is consistent with the earlier result which states that the system has a unique massive phase, which interpolates smoothly between the classical weakly-coupled limit for LΛ → 0 and the “confined” phase of the standard $$ \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} $$ model in two dimensions for LΛ → ∞.