Three dimensional conformal sigma models

TL;DR

This paper constructs three-dimensional conformal sigma models using Wilsonian renormalization group methods, identifying fixed points and characterizing target space properties based on anomalous dimensions.

Contribution

It introduces a novel approach to define conformal sigma models in three dimensions via fixed points of the Wilsonian RG, linking them to Ricci flow and Einstein-Kähler manifolds.

Findings

Existence of fixed points in 3D sigma models confirmed.

Target space geometry varies with anomalous dimension.

Conformal models correspond to Einstein-Kähler manifolds at specific parameters.

Abstract

We construct novel conformal sigma models in three dimensions. Nonlinear sigma models in three dimensions are nonrenormalizable in perturbation theory. We use Wilsonian renormalization group equation method to find the fixed points. Existence of fixed points is extremely important in this approach to show the renormalizability. Conformal sigma models are defined as the fixed point theories of the Wilsonian renormalization group equation. The Wilsonian renormalization group equation with anomalous dimension coincides with the modified Ricci flow equation. The conformal sigma models are characterized by one parameter which corresponds to the anomalous dimension of the scalar fields. Any Einstein-K\"{a}hler manifold corresponds to a conformal field theory when the anomalous dimension is $\gamma=-1/2$. Furthermore, we investigate the properties of target spaces in detail for two dimensional case, and find the target space of the fixed point theory becomes compact or noncompact depending on the value of the anomalous dimension.