Noncommutative (generalized) sine-Gordon/massive Thirring correspondence, integrability and solitons

Abstract

Some properties of the correspondence between the non-commutative versions of the (generalized) sine-Gordon (NCGSG1,2) and the massive Thirring (NCGMT1,2) models are studied. Our method relies on 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 (NCATM1,2), in which the Toda field g belongs to certain subgroups of GL(3), and the matter fields lie in the higher grading directions of an affine Lie algebra. Depending on the form of g one arrives at two different NC versions of the NCGSG1,2/NCGMT1,2 correspondence. In the NCGSG1,2 sectors, through consistent reduction procedures, we find NC versions of some well-known models, such as the NC sine-Gordon (NCSG1,2) (Lechtenfeld et al and Grisaru-Penati proposals, respectively), NC (bosonized) Bukhvostov-Lipatov (NCbBL1,2) and NC double sine-Gordon (NCDSG1,2) models. The NCGMT1,2 models correspond to Moyal product extension of the generalized massive Thirring model. The NCGMT1 ,2 models posses constrained versions with relevant Lax pair formulations, and other sub-models such as the NC massive Thirring (NCMT1,2), the NC Bukhvostov-Lipatov (NCBL1,2) and constrained versions of the last models with Lax pair formulations. We have established that, except for the well known NCMT1,2 zero-curvature formulations, generalizations (nF ≥ 2, nF =number of flavors) of the massive Thirring model allow zero-curvature formulations only for constrained versions of the models and for each one of the various constrained sub-models defined for less than nF flavors, in the both NCGMT1,2 and ordinary space-time descriptions (GMT), respectively. The non-commutative solitons and kinks of the GL(3) NCGSG1,2 models are investigated.