Three Dimensional Nonlinear Sigma Models in the Wilsonian Renormalization Method

TL;DR

This paper demonstrates that three-dimensional supersymmetric nonlinear sigma models can be renormalizable using the nonperturbative Wilsonian renormalization group, identifying UV fixed points for certain target space geometries and constructing related conformal field theories.

Contribution

It shows the renormalizability of 3D supersymmetric sigma models via nonperturbative methods and constructs new models with multiple UV fixed points and conformal field theories.

Findings

Positive curvature target spaces have nontrivial UV fixed points.

Negative curvature target spaces lack nontrivial UV fixed points.

Interpolating models exhibit multiple UV fixed points for continuum limits.

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 ${\cal N}=2$ supersymmetric nonlinear $\sigma$ models are renormalizable in three dimensions. When the target space is an Einstein-K\"{a}hler 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 ${\bf SU}(N)$ 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 ${\bf SU}(N+1)$.