Integrability of the $\lambda$-deformation of the PCM with spectators

TL;DR

This paper generalizes the $mbda$-deformation of the Principal Chiral Model by deforming a subgroup, revealing conditions for integrability and analyzing the resulting Lax connection and RG flow.

Contribution

It introduces a new class of $mbda$-deformations with a symmetric coset restriction and uncovers a novel pole structure in the Lax connection.

Findings

Integrability requires the coset $F\backslash G$ to be symmetric.

The Lax connection exhibits four poles, unlike previous models.

Explicit RG flow analysis for $G=SU(2)$, $F=U(1)$.

Abstract

We construct a generalisation of the $\lambda$-deformation of the Principal Chiral Model (PCM) where we deform just a subgroup $F$ of the full symmetry group $G$. We find that demanding Lax integrability imposes a crucial restriction, namely that the coset $F\backslash G$ must be symmetric. Surprisingly, we also find that (when $F$ is non-abelian) integrability requires that the term in the action involving only the spectator fields should have a specific $\lambda$-dependence, which is a curious modification of the procedure expected from the known $F=G$ case. The resulting Lax connection has a novel analytical structure, with four single poles as opposed to the two poles of the cases of the PCM and of the standard $\lambda$-deformation. We also explicitly work out the example of $G=SU(2)$ and $F=U(1)$, discussing its renormalisation group flow to two loops.