Field theory and $\lambda$-deformations: Expanding around the identity

We explore the structure of the $\lambda$-deformed $\sigma$-model action by setting up a perturbative expansion around the free field point corresponding to the identity group element. We include all field interaction terms up to sixth order. We compute the two- and three-point functions of current and primary filed operators, their anomalous dimensions as well as the $\beta$-function for the $\lambda$-parameter. Our results are in complete agreement with those obtained previously using gravitational and/or CFT perturbative methods in conjunction with the non-perturbative symmetry, as well as with those obtained using methods exploiting the geometry defined in the space of couplings. The advantage of this approach is that all deformation effects are already encoded in the couplings of the interaction vertices and in the $\lambda$-dressed operators.