Integrability and RG flow in 2d sigma models

TL;DR

This paper investigates the stability of integrable 2D sigma models under quantum RG flow, constructs quantum corrections for specific models, and explores their reformulations and new classes, linking integrability with RG flow and string theory applications.

Contribution

It demonstrates that integrable sigma models remain RG-stable at higher loops with specific quantum corrections and introduces new integrable models and reformulations, expanding understanding of solvable string theories.

Findings

Integrable sigma models are RG-stable at higher loops with appropriate quantum corrections.

Explicit quantum corrections are constructed for eta- and lambda-deformed models.

New classes of integrable models with local couplings depending on 2D time are proposed.

Abstract

Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d $\sigma$-models. We include non-conformal $\sigma$-models, which have historically been a good arena for discovering integrable models that were later generalized to Weyl-invariant ones. General $\sigma$-models feature a quantum RG flow, given by a 'generalized Ricci flow' of the target-space geometry. This thesis is based on the conjecture that integrable $\sigma$-models are renormalizable, or stable under the RG flow. It is widely understood that classically integrable theories are stable at the leading 1-loop order with only a few parameters running. Here we address what happens at higher-loop orders. We find that integrable $\sigma$-models generally remain RG-stable at higher-loops provided they receive a particular choice of finite counterterms, or quantum ($\alpha'$) corrections to the target-space geometry. We explicitly construct these quantum corrections for examples of integrable $\eta$- and $\lambda$-deformed $\sigma$-models. We then reformulate the $\lambda$-models as $\sigma$-models on a "tripled" $G \times G \times G$ configuration space, where they become automatically renormalizable due to manifest symmetries and a decoupling of some fields. We also consider the integrable $G \times G$ and $G \times G/H$ models and construct a new class of integrable $G \times G/H$ models with abelian $H$. We then present a new and different link between integrability and the RG flow in the context of $\sigma$-models with 'local couplings' depending explicitly on 2d time. Such models are naturally obtained in the light-cone gauge in string theory, pointing to the possibility of a large, new class of solvable string models.