Homogeneous Yang-Baxter deformations as undeformed yet twisted models

TL;DR

This paper demonstrates that homogeneous Yang-Baxter deformations of integrable models are equivalent to undeformed models with twisted boundary conditions, providing a local expression for the twist and analyzing its implications for spectral data.

Contribution

The authors derive a local expression for the twist in homogeneous Yang-Baxter deformations, enabling practical calculations and analysis of spectral curves.

Findings

The twist can be expressed locally in terms of undeformed model variables.

The local twist expression simplifies the construction of the classical spectral curve.

Analysis of specific deformations illustrates the impact on monodromy and spectral data.

Abstract

The homogeneous Yang-Baxter deformation is part of a larger web of integrable deformations and dualities that recently have been studied with motivations in integrable $\sigma$-models, solution-generating techniques in supergravity and Double Field Theory, and possible generalisations of the AdS/CFT correspondence. The $\sigma$-models obtained by the homogeneous Yang-Baxter deformation with periodic boundary conditions on the worldsheet are on-shell equivalent to undeformed models, yet with twisted boundary conditions. While this has been known for some time, the expression provided so far for the twist features non-localities (in terms of the degrees of freedom of the deformed model) that prevent practical calculations, and in particular the construction of the classical spectral curve. We solve this problem by rewriting the equation defining the twist in terms of the degrees of freedom of the undeformed yet twisted model, and we show that we are able to solve it in full generality. Remarkably, this solution is a local expression. We discuss the consequences of the twist at the level of the monodromy matrix and of the classical spectral curve, analysing in particular the concrete examples of abelian, almost abelian and Jordanian deformations of the Yang-Baxter class.