Affine Toda systems coupled to matter fields

Abstract

We investigate higher grading integrable generalizations of the affine Toda systems, where the flat connections defining the models take values in eigensubspaces of an integral gradation of an affine Kac-Moody algebra, with grades varying from l to −l (l > 1). The corresponding target space possesses nontrivial vacua and soliton configurations, which can be interpreted as particles of the theory, on the same footing as those associated to fundamental fields. The models can also be formulated by a Hamiltonian reduction procedure from the so called two-loop WZNW models. We construct the general solution and show the classes corresponding to the solitons. Some of the particles and solitons become massive when the conformal symmetry is spontaneously broken by a mechanism with an intriguing topological character and leading to a very simple mass formula. The massive fields associated to non zero grade generators obey field equations of the Dirac type and may be regarded as matter fields. A special class of models is remarkable. These theories possess a U (1) Noether current which, after a special gauge fixing of the conformal symmetry, is proportional to a topological current. This leads to the confinement of the matter field inside the solitons, which can be regarded as a one dimensional bag model for QCD. These models are also relevent to the study of electron self-localization in (quasi)-one-dimensional electron-phonon systems.

Sine–Gordon Model in the Homogeneous Higher Grading

“…In this subsection we recall some facts about affine Kac-Moody algebras [7], [4]. Consider an untwisted affine Kac-Moody algebra G endowed with an integral grading G = n∈Z Z G n , and denote G ± = n>0 G ±n .…”

“…Applying the zero-curvature conditions on elements of connection containing Lie algebra generators in appropriate grading subspaces, we obtain systems of equations of motion associated to a specific Lie algebra. In [4] (of which we keep notations) the higher grading generalization to the conformal affine Toda models was considered. Elements of the higher (then number one) grading subspaces are taking into account while connection elements are constructed.…”

“…The case l = 1 Now consider the case l = 1. Let us parameterize the element B in the homogeneous grading of G, [4]. From the equations (6-9) for an infinite dimensional Lie algebra G in the principal grading we obtain the affine Toda field theory systems of equations…”

“…The element B is parameterized as B = e ϕ·H 0 e ν C e ηQ ppal = e ϕ·H 0 eν C e ηQ ppal , wherẽ H 0 was defined in [4] …”

“…In [4] the higher grading generalization to the conformal affine Toda models was considered. Elements of the higher (then number one) grading subspaces are taking into account while connection elements are constructed.…”