Non-commutative solitons and strong-weak duality

TL;DR

This paper explores non-commutative extensions of sine-Gordon and Thirring models using integrable systems and master Lagrangians, revealing soliton solutions and dualities that may persist quantum mechanically.

Contribution

It introduces a novel approach to non-commutative integrable models via master Lagrangians, establishing classical soliton solutions and dualities in NC sine-Gordon and Thirring theories.

Findings

Exact one-soliton solutions found without NC parameter expansion

Identified strong-weak duality between NC sine-Gordon and Thirring models

Models share common soliton solutions, indicating deep duality

Abstract

Some properties of the non-commutative versions of the sine-Gordon model (NCSG) and the corresponding massive Thirring theories (NCMT) are studied. Our method relies on the NC extension of integrable models and the master Lagrangian approach to deal with dual theories. The master Lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM) associated to the group GL(2), in which the Toda field belongs to certain representations of either $U(1){x} U(1)$ or $U(1)_{C}$ corresponding to the Lechtenfeld et al. (NCSG$_{1}$) or Grisaru-Penati (NCSG$_{2}$) proposals for the NC versions of the sine-Gordon model, respectively. Besides, the relevant NCMT$_{1, 2}$ models are written for two (four) types of Dirac fields corresponding to the Moyal product extension of one (two) copy(ies) of the ordinary massive Thirring model. The NCATM$_{1,2}$ models share the same one-soliton (real Toda field sector of model 2) exact solutions, which are found without expansion in the NC parameter $\theta$ for the corresponding Toda and matter fields describing the strong-weak phases, respectively. The correspondence NCSG$_{1}$ $\leftrightarrow$ NCMT$_{1}$ is promising since it is expected to hold on the quantum level.