Ultralocal Lax connection for para-complex $\mathbb{Z}_T$-cosets

TL;DR

This paper demonstrates the existence of an ultralocal gauge-invariant Lax connection for sigma-models on para-complex $ ext{Z}_T$-cosets, extending previous results to new non-hermitian symmetric spaces with ultralocal Poisson brackets.

Contribution

It introduces a novel ultralocal Lax connection for para-complex $ ext{Z}_T$-cosets, broadening the class of integrable models with ultralocal Poisson structures.

Findings

Existence of gauge-invariant Lax connection with ultralocal Poisson brackets.

Light-cone components of the Lax connection commute under Poisson brackets.

Extension of previous hermitian symmetric space results to para-complex $ ext{Z}_T$-cosets.

Abstract

We consider $\sigma$-models on para-complex $\mathbb{Z}_T$-cosets, which are analogues of those on complex homogeneous target spaces considered recently by D. Bykov. For these models, we show the existence of a gauge-invariant Lax connection whose Poisson brackets are ultralocal. Furthermore, its light-cone components commute with one another in the sense of Poisson brackets. This extends a result of O. Brodbeck and M. Zagermann obtained twenty years ago for hermitian symmetric spaces.