Higher grading conformal affine Toda theory and (generalized) sine-Gordon/massive Thirring duality

Abstract

Some properties of the higher grading integrable generalizations of the conformal affine Toda systems are studied. The fields associated to the non-zero grade generators are Dirac spinors. The effective action is written in terms of the Wess-Zumino-Novikov-Witten (WZNW) action associated to an affine Lie algebra, and an off-critical theory is obtained as the result of the spontaneous breakdown of the conformal symmetry. Moreover, the off-critical theory presents a remarkable equivalence between the Noether and topological currents of the model. Related to the off-critical model we define a real and local Lagrangian provided some reality conditions are imposed on the fields of the model. This real action model is expected to describe the soliton sector of the original model, and turns out to be the master action from which we uncover the weak-strong phases described by (generalized) massive Thirring and sine-Gordon type models, respectively. The case of any (untwisted) affine Lie algebra furnished with the principal gradation is studied in some detail. The example ofŝl(n) (n=2,3) is presented explicitly.

Non-commutative solitons and strong-weak duality

“…The study of these models become interesting since the su(n) ATM theories constitute excellent laboratories to test ideas about confinement [10,12], the role of solitons in quantum field theories [7], duality transformations interchanging solitons and particles [7,13], as well as the reduction processes of the (two-loop) Wess-Zumino-Novikov-Witten (WZNW) theory from which the ATM models are derivable [14,9]. Moreover, the ATM type systems may also describe some low dimensional condensed matter phenomena, such as self-trapping of electrons into solitons, see e.g.…”

“…In order to derive the NC versions of the sine-Gordon model we follow the master lagrangian approach [18], as in the ordinary SG derivation [9], starting from the NCATM 1,2 models (2.1) and (2.11), respectively. Let us concentrate first on the equations of motion (2.3)-(2.5) which are understood to be written for g ∈ U(1) × U(1) or U(1) C .…”

“…In the study of the ordinary (commutative) ATM model performed in [7]- [9], the massive Thirring model (MT) was obtained by means of a hamiltonian reduction and the so-called decoupling procedures. The first procedure requires the definition of conjugated momenta for the fields of the model.…”

New derivation of soliton solutions to the AKNS2system via dressing transformation methods

“…The dark solitons, as in section 5, will require the vertex operator of type W q (k, ρ ± j ) (j = 1, 2, ..., r), the analog of the operator in (C.7) incorporating additional terms. Finally, the mixed boundary conditions and the dark-bright solitons will emerge by extending the discussion in section (6). In the case of the vector 1-soliton solution it is possible to form the combination (m, r − m), m =number of dark components, r − m =number of bright components.…”

“…The model defined by two coupled NLS systems was earlier studied by Manakov [5]. Another remarkable example of a multi-field generalization of an integrable model is the so-called generalized sine-Gordon model which is integrable in some regions of its parameter space and it has many physical applications [6,7]. The type of coupled NLS equations find applications in diverse areas of physics such as non-linear optics, optical communication, biophysics, multi-component Bose-Einsten condensate, etc (see e.g.…”