The Moyal Momentum algebra applied to θ - deformed 2 d conformal models and KdV - hierarchies

The properties of the Das-Popowicz Moyal momentum algebra that we introduce in hep-th/0207242 are reexamined in details and used to discuss some aspects of integrable models and 2d conformal field theories. Among the results presented we setup some useful convention notations which lead to extract some non trivial properties of the Moyal momentum algebra. We use the particular sub-algebra sln−Σ̂ (0,n) n to construct the sl2Liouville conformal model ∂∂̄φ = 2 θ e 1 θ φ and its sl3-Toda extension ∂∂̄φ1 = Ae 1 2θ (φ1+ 1 2 φ2) and ∂∂̄φ2 = Be − 1 2θ 12. We show also that the central charge, a la Feigin-Fuchs, associated to the spin-2 conformal current of the θ-Liouville model is given by cθ = (1+24θ 2). Moreover, the results obtained for the Das-Popowicz Mm algebra are applied to study systematically some properties of the Moyal KdV and Boussinesq hierarchies generalizing some known results. We discuss also the primarity condition of conformal wθcurrents and interpret this condition as being a dressing gauge symmetry in the Moyal momentum space. Some computations related to the dressing gauge group are explicitly presented. Corresponding author: [email protected].