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 S O ( 1 , 2 ) / S O ( 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.