Classical solutions of sigma models in curved backgrounds by the Poisson-Lie T-plurality

TL;DR

This paper demonstrates how Poisson-Lie T-plurality transformations can convert complex sigma-model equations in curved backgrounds into simpler flat background equations, enabling explicit solutions and preserving string conditions.

Contribution

It introduces a method to explicitly solve sigma-model equations in curved backgrounds using Poisson-Lie T-plurality transformations, including coordinate transformations and PDE solutions.

Findings

Explicit solutions for sigma-models in curved backgrounds

Preservation of string conditions under Poisson-Lie transformations

Application to three-dimensional relativistic string equations

Abstract

Classical equations of motion for three-dimensional sigma-models in curved background are solved by a transformation that follows from the Poisson-Lie T-plurality and transform them into the equations in the flat background. Transformations of coordinates that make the metric constant are found and used for solving the flat model. The Poisson-Lie transformation is explicitly performed by solving the PDE's for auxiliary functions and finding the relevant transformation of coordinates in the Drinfel'd double. String conditions for the solutions are preserved by the Poisson-Lie transformations. Therefore we are able to specify the type of sigma-model solutions that solve also equations of motion of three dimensional relativistic strings in the curved backgrounds. Simple examples are given.