Lagrangian Formulation of Symmetric Space sine-Gordon Models

TL;DR

The paper addresses the problem of obtaining a Lagrangian description for symmetric space sine-Gordon (SSSG) models arising from conformal reduction of 2D sigma-models. It introduces aunifying approach based on a triplet of groups F ⊇ G ⊇ H, formulating the action as a gauged WZW model for the coset G/H augmented by a potential I_P(g,T,Tbar), and demonstrates a zero-curvature (Lax) representation that preserves integrability. The authors derive explicit vector-type SSSG equations for several compact type I spaces (e.g., $SO(n+1)/SO(n)$, $SU(n)/SO(n)$, $SU(n+1)/U(n)$, and $Sp(n)/U(n)$), and construct Backlund transformations and soliton solutions, including a detailed 1-soliton in $SU(3)/SO(3)$. They show that SSSG models are integrable perturbations of coset CFTs at the classical level and discuss vacua, Backlund structure, and generalizations, paving the way for a quantum perturbative treatment and connections to non-abelian parafermions and SL(2) embeddings.

Abstract

The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim $σ$-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by considering a triplet of Lie groups $F \supset G \supset H$. We show that for every symmetric space $F/G$, the generalized sine-Gordon models can be derived from the $G/H$ WZW action, plus a potential term that is algebraically specified. Thus, the symmetric space sine-Gordon models describe certain integrable perturbations of coset conformal field theories at the classical level. We also briefly discuss their vacuum structure, Backlund transformations, and soliton solutions.