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

## TL;DR

This paper investigates quantum corrections to classically integrable 2D sigma models, revealing the necessity of counterterms at higher loops to preserve integrability and RG flow properties, with detailed examples and implications for duality.

## Contribution

It demonstrates that quantum counterterms are essential for maintaining integrability and RG flow consistency at higher loops in certain 2D sigma models, extending classical results.

## Key findings

- Quantum counterterms are needed at 2-loop order to preserve integrability.
- Examples include the eta-deformation of S^2 and H^2, and the lambda-deformation of SO(1,2)/SO(2).
- Counterterms are also necessary for non-abelian duality to commute with RG flow.

## Abstract

Classically integrable $\sigma$-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 $\sigma$-model should be corrected by quantum counterterms. The pattern is similar to that of effective $\sigma$-models associated to gauged WZW theories. We consider in detail the examples of the $\eta$-deformation of $S^2$ ("sausage model") and $H^2$, as well as the closely related $\lambda$-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.