Pohlmeyer reduction revisited

TL;DR

This paper revisits the Pohlmeyer reduction, providing a systematic group theoretical framework that links sigma models on symmetric spaces to integrable equations, with applications to classical string configurations in various curved space-times.

Contribution

It offers a comprehensive group theoretical formulation of Pohlmeyer reduction, clarifies conditions for Lagrangian formulations, and discusses specific examples like CP^n and AdS_n.

Findings

Established a map between sigma models and integrable equations.

Clarified conditions for Lagrangian formulations with gauged WZW actions.

Analyzed Pohlmeyer reductions for CP^n and AdS_n spaces.

Abstract

A systematic group theoretical formulation of the Pohlmeyer reduction is presented. It provides a map between the equations of motion of sigma models with target-space a symmetric space M=F/G and a class of integrable multi-component generalizations of the sine-Gordon equation. When M is of definite signature their solutions describe classical bosonic string configurations on the curved space-time R_t\times M. In contrast, if M is of indefinite signature the solutions to those equations can describe bosonic string configurations on R_t\times M, M\times S^1_\vartheta or simply M. The conditions required to enable the Lagrangian formulation of the resulting equations in terms of gauged WZW actions with a potential term are clarified, and it is shown that the corresponding Lagrangian action is not unique in general. The Pohlmeyer reductions of sigma models on CP^n and AdS_n are discussed as particular examples of symmetric spaces of definite and indefinite signature, respectively.