RG flows for $\lambda$-deformed CFTs

We study the renormalization group equations of the fully anisotropic λ -deformed CFTs involving the direct product of two current algebras at different levels k1,2 for general semi-simple groups. The exact, in the deformation parameters, β -function is found via the effective action of the quantum fluctuations around a classical background as well as from gravitational techniques. Furthermore, agreement with known results for symmetric couplings and/or for equal levels, is demonstrated. We study in detail the two coupling case arising by splitting the group into a subgroup and the corresponding coset manifold which consistency requires to be either a symmetric-space one or a non-symmetric Einstein-space.