Integrability of the λ-deformation of the PCM with spectators

We construct a generalisation of the λ-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 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 λ-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 λ-deformation. We also explicitly work out the example of $G=SU(2)$ and $F=U(1)$, discussing its renormalisation group flow to two loops.