Sigma Models as Perturbed Conformal Field Theories

We show that two-dimensional sigma models are related to certain perturbed conformal field theories. When the fields in the sigma model take values in a space $G/H$ for a group $G$ and a maximal subgroup $H$, we argue that the corresponding conformal field theory is the $k\to \infty$ limit of the coset model $(G/H{)}_k$, and the perturbation is related to the current of $G$. Nonperturbative instanton contributions to the sigma model free energy are perturbative when $k$ is finite. We use this mapping to find the free energy for the “ $\mathrm{O}\left(n\right)$” [ $=\mathrm{O}\left(n\right)/\mathrm{O}(n-1\mathrm{})$] sigma model at nonzero temperature. It also results in a new approach to the ${\mathrm{CP}}^n$ model.