RG flow of integrable $\mathcal{E}$-models

TL;DR

This paper analyzes the renormalization group flow of integrable sigma-models with Poisson-Lie symmetry, revealing their one-loop renormalizability and providing explicit flow equations, with applications to known deformations.

Contribution

It introduces a unified geometric framework for integrable deformations via twist functions and computes their RG flow at one and two loops, highlighting conditions for renormalizability.

Findings

Models are one-loop renormalisable with simple flow expressions.

Two-loop renormalisability only for models with N=1.

Results reproduce known beta-functions for lambda-deformations.

Abstract

We compute the one- and two-loop RG flow of integrable $\sigma$-models with Poisson-Lie symmetry. They are characterised by a twist function with $2N$ simple poles/zeros and a double pole at infinity. Hence, they capture many of the known integrable deformations in a unified framework, which has a geometric interpretation in terms of surface defects in a 4D Chern-Simons theory. We find that these models are one-loop renormalisable and present a very simple expression for the flow of the twist function. At two loops only models with $N$=1 are renormalisable. Applied to the $\lambda$-deformation on a semisimple group manifold, our results reproduce the $\beta$-functions in the literature.