New integrable coset sigma models

TL;DR

This paper introduces a new class of integrable sigma models based on affine Gaudin models, generalizing symmetric space models, and explores their Lagrangian formulations and specific examples with novel target spaces.

Contribution

The authors construct a new family of integrable sigma models on cosets of multiple Lie groups, derive their Lagrangians, and identify a novel three-parametric model with target space T^{1,1}.

Findings

Constructed a new class of integrable sigma models using affine Gaudin models.

Derived explicit Lagrangian for the N=2 case showing a simple form related to the classical R-matrix.

Identified a new three-parametric integrable model with T^{1,1} target space.

Abstract

By using the general framework of affine Gaudin models, we construct a new class of integrable sigma models. They are defined on a coset of the direct product of $N$ copies of a Lie group over some diagonal subgroup and they depend on $3N-2$ free parameters. For $N=1$ the corresponding model coincides with the well-known symmetric space sigma model. Starting from the Hamiltonian formulation, we derive the Lagrangian for the $N=2$ case and show that it admits a remarkably simple form in terms of the classical $\mathcal{R}$-matrix underlying the integrability of these models. We conjecture that a similar form of the Lagrangian holds for arbitrary $N$. Specifying our general construction to the case of $SU(2)$ and $N=2$, and eliminating one of the parameters, we find a new three-parametric integrable model with the manifold $T^{1,1}$ as its target space. We further comment on the connection of our results with those existing in the literature.