The Moyal Momentum algebra applied to (theta)-deformed 2d conformal models and KdV-hierarchies

TL;DR

This paper explores the properties of the Moyal momentum algebra in 2D conformal models and KdV hierarchies, revealing new algebraic structures, conformal models, and hierarchies with generalized results.

Contribution

It introduces detailed properties of the Moyal momentum algebra and applies them to construct conformal models and hierarchies, extending known results in integrable systems.

Findings

Derived the central charge c(θ)=1+24θ^2 for the θ-Liouville model.

Constructed sl(2)-Liouville and sl(3)-Toda models using the Moyal algebra.

Generalized KdV and Boussinesq hierarchies within the Moyal framework.

Abstract

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 sl(n)-{Sigma}_{n}^{(0,n)} to construct the sl(2)-Liouville conformal model and its sl(3)-Toda extension. We show also that the central charge, a la Feigin-Fuchs, associated to the spin-2 conformal current of the (theta)-Liouville model is given by c(theta)=1+24.theta^{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 $w_{\theta}$-currents 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.