Supersymmetric three dimensional conformal sigma models

TL;DR

This paper constructs three-dimensional supersymmetric conformal sigma models using the Wilsonian renormalization group, identifying fixed points and their properties related to target space geometry and anomalous dimensions.

Contribution

It introduces a nonperturbative approach to find fixed points of 3D supersymmetric sigma models via the Wilsonian RG, linking conformal invariance to target space geometry.

Findings

Existence of fixed points characterized by anomalous dimensions.

Target space geometry depends on the anomalous dimension value.

Conformal sigma models correspond to Einstein-Kähler manifolds at specific anomalous dimensions.

Abstract

We construct supersymmetric conformal sigma models in three dimensions. Nonlinear sigma models in three dimensions are nonrenormalizable in perturbation theory. We use the Wilsonian renormalization group equation method, which is one of the nonperturbative methods, 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. This is a proceeding to the conference based on the talk given by E.I.