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

Abstract

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 S2 (“sausage model”) and H2, 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.