All-loop anomalous dimensions in integrable $\lambda$-deformed $\sigma$-models

TL;DR

This paper computes the all-loop anomalous dimensions of current operators in $ ext{lambda}$-deformed $ ext{sigma}$-models using multiple methods, advancing understanding of their quantum properties and symmetries.

Contribution

It provides the first comprehensive calculation of all-loop anomalous dimensions in $ ext{lambda}$-deformed models for various groups and deformations, employing effective action, perturbation theory, and CFT techniques.

Findings

Derived all-loop anomalous dimensions for isotropic and anisotropic cases.

Established duality symmetry relations in the models.

Extended results to specific groups like SU(2) and coset spaces.

Abstract

We calculate the all-loop anomalous dimensions of current operators in $\lambda$-deformed $\sigma$-models. For the isotropic integrable deformation and for a semi-simple group $G$ we compute the anomalous dimensions using two different methods. In the first we use the all-loop effective action and in the second we employ perturbation theory along with the Callan-Symanzik equation and in conjunction with a duality-type symmetry shared by these models. Furthermore, using CFT techniques we compute the all-loop anomalous dimensions of bilinear currents for the isotropic deformation case and a general $G$. Finally we work out the cases of anisotropic $SU(2)$ and the two coupling, corresponding to the symmetric coset $G/H$ and a subgroup $H$, splitting of a group $G$.