Large-N Analysis of Three Dimensional Nonlinear Sigma Models

Abstract

Non-perturbative renormalization group approach suggests that a large class of nonlinear sigma models are renormalizable in three dimensional space-time, while they are non-renormalizable in perturbation theory. ${\cal N}=2$ supersymmetric nonlinear sigma models whose target spaces are Einstein-Kähler manifolds with positive scalar curvature belongs to this class. hermitian symmetric spaces, being homogeneous, are specially simple examples of these manifolds. To find an independent evidence of the nonperturbative renormalizability of these models, the large N method, another nonperturbative method, is applied to 3-dimensional ${\cal N}=2$ supersymmetric nonlinear sigma models on the target spaces $CP^{N-1}=SU(N)/[SU(N-1)\times U(1)]$ and $Q^{N-2}=SO(N)/[SO(N-2)\times SO(2)]$, two typical examples of hermitian symmetric spaces. We find that $β$ functions in these models agree with the results of the nonperturbative renormalization group approach in the next-to-leading order of 1/N expansion, and have non-trivial UV fixed points. The $β$ function of the $Q^{N-2}$ model receives a nonzero correction in the next-to-leading order of the 1/N expansion. We also investigate the phase structures of our models. The $CP^{N-1}$ model has two phases; SU(N) symmetric and asymmetric phase. The $Q^{N-2}$ model has three phases; Chern-Simons, Higgs and SO(N) broken phases. In the Chern-Simons and Higgs phase, SO(N) symmetry remains unbroken and all dynamical fields becomes massive. An auxiliary gauge field also acquires mass, through an induced Chern-Simons term in the Chern-Simons phase, and through the vacuum expectation value of a di-quark bound state in the Higgs phase.