Integrability vs. RG flow in $G \times G$ and $G \times G /H$ sigma models

TL;DR

This paper explores the relationship between integrability and RG flow stability in certain 2D sigma models based on product group spaces, demonstrating that integrability can be preserved under 2-loop RG flow and constructing new integrable models.

Contribution

It shows that integrability in $G imes G$ models persists under 2-loop RG flow and introduces new integrable $G imes G /H$ models, including a novel $T^{1,q}$ space model.

Findings

Integrability is preserved under 2-loop RG flow in $G imes G$ models.

New integrable $G imes G /H$ models are constructed with abelian subgroup $H$.

The $T^{1,q}$ model is stable under RG flow and related to T-duality.

Abstract

We consider a class of 2d $\sigma$-models on products of group spaces that provide new examples of a close connection between integrability and stability under the RG flow. We first study the integrable $G \times G$ model derived from the affine Gaudin construction (for which the 1-loop $\beta$-functions were found in arXiv:2010.07879) and show that its condition of integrability is preserved also by the 2-loop RG flow. We then investigate the RG flow in the gauged $G \times G /H$ model, in particular the integrable $T^{1,1}$ model found in arXiv:2010.05573. We also construct a new class of integrable $G \times G /H$ models in the case when the subgroup $H$ is abelian. In the simplest case of $G=SU_2$, $H=U_1$ this leads to an integrable $\sigma$-model on the $T^{1,q}$ space (with a particular $B$-field). This model is also shown to be stable under the 2-loop RG flow, and we relate this property to its invariance under T-duality in an isometric $U_1$ direction. This $T^{1,q}$ model may be interpreted as an integrable deformation of the GMM model (of two coupled WZW theories with generic levels) away from the conformal point.