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

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 $G$.