Three Dimensional Nonlinear Sigma Models in the Wilsonian Renormalization Method

Abstract

The three dimensional nonlinear sigma model is unrenormalizable in perturbative method. By using the $\beta$ function in the nonperturbative Wilsonian renormalization group method, we argue that $\mathcal{N}=2$ supersymmetric nonlinear $\sigma$ models are renormalizable in three dimensions. When the target space is an Einstein-Kähler manifold with positive scalar curvature, such as C$P^N$ or $Q^N$, there are nontrivial ultraviolet (UV) fixed point, which can be used to define the nontrivial continuum theory. If the target space has a negative scalar curvature, however, the theory has only the infrared Gaussian fixed point, and the sensible continuum theory cannot be defined. We also construct a model which interpolates between the C$P^N$ and $Q^N$ models with two coupling constants. This model has two non-trivial UV fixed points which can be used to define the continuum theory. Finally, we construct a class of conformal field theories with $\mathbf{SU}(N)$ symmetry, defined at the fixed point of the nonperturbative $\beta$ function. These conformal field theories have a free parameter corresponding to the anomalous dimension of the scalar fields. If we choose a specific value of the parameter, we recover the conformal field theory defined at the UV fixed point of C$P^N$ model and the symmetry is enhanced to $\mathbf{SU}(N+1)$.

16 pages, 1 figure, references added

Cited by in corpus (7)

• Fixed Points of Nonlinear Sigma Models in d>2
• Conformal Sigma Models with Anomalous Dimensions and Ricci Solitons
• On the Renormalization of Theories of a Scalar Chiral Superfield
• Auxiliary Field Methods in Supersymmetric Nonlinear Sigma Models
• Deformation of Curved BPS Domain Walls and Supersymmetric Flows on 2d Kähler-Ricci Soliton
• All orders analysis of three dimensional CP^(N-1) model in 1/N-expansion
• Supersymmetric three dimensional conformal sigma models