Singular Non-Abelian Toda Theories

The algebraic conditions that specific gaugedG/H-WZWmodel have to satisfy in order to give rise to Non-Abelian Toda models with non singular metric with or without torsion are found. The classical algebras of symmetries corresponding to grade one rank 2 and 3 singular NA-Toda models are derived. The proliferation of different two dimensional G0-Toda theories ([2] [10] ) addresses the question about their algebraic classification in terms of a G0 ⊂ G embedding and a G-invariant WZNW -model. The simplest class of models is the Abelian Toda, which is known to be completely integrable and conformal invariant. These models correspond to a Abelian subgroup G0 ⊂ G and their symmetries generate the Wn-algebra (n=rank G). The non-Abelian Toda models, in turn are connected to non-abelian embeddings G0 ⊂ G and describe a string propagating on a specific curved background, containing also tachyons, dilatons Φ(X) and, possibly axions. Its general action is of the form (i, j = 1, · · · , D; μ, ν = 0, 1): S = ∫ dz{(Gij(X)ημν + ǫμνBij(X)) ∂μX ∂νX − α′R(2)Φ(X) + Tach. potential}. The properties of the background metric Gij and of the anti-symmetric term Bij(torsion) depend upon the embedding G0 ⊂ G, whereG0 is now non-Abelian and classify the models according to singular or non-singular metrics and the presence of axionic or torsionless terms. The symmetries of the singular metric NA-Toda models are described by a non-local algebra, which corresponds to the semi-classical limit of a mixed parafermionic and W -algebra structure, denoted by V -algebra (see [3], [2], [4] and [5]). On leave of absence from the Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784, Sofia