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Integrable 2d sigma models: quantum corrections to geometry from RG flow

Classically integrable $σ$-models are known to be solutions of the 1-loop RG equations, or "Ricci flow", with only a few couplings running. In some of the simplest examples of integrable deformations we find that in order to preserve this property at 2 (and higher) loops the classical $σ$-model should be corrected by quantum counterterms. The pattern is similar to that of effective $σ$-models associated to gauged WZW theories. We consider in detail the examples of the $η$-deformation of $S^2$ ("sausage model") and $H^2$, as well as the closely related $λ$-deformation of the $SO(1,2)/SO(2)$ coset. We also point out that similar counterterms are required in order for non-abelian duality to commute with RG flow beyond the 1-loop order.