Dressing Operator Approach to Moyal Algebraic Deformation of Selfdual Gravity

Recently Strachan introduced a Moyal algebraic deformation of selfdual gravity, replacing a Poisson bracket of the Plebanski equation by a Moyal bracket. The dressing operator method in soliton theory can be extended to this Moyal algebraic deformation of selfdual gravity. Dressing operators are defined as Laurent series with coefficients in the Moyal (or star product) algebra, and turn out to satisfy a factorization relation similar to the case of the KP and Toda hierarchies. It is a loop algebra of the Moyal algebra (i.e., of a W∞ algebra) and an associated loop group that characterize this factorization relation. The nonlinear problem is linearized on this loop group and turns out to be integrable. The notion of Moyal algebras [1] is a kind of quantum deformation of Poisson algebras. Since the end of the eighties, there has been renewed interest in these algebras. This is because in two dimensions, they give an explicit realization of the two types of W-infinity algebras — quantum (W∞) and classical (w∞) algebras. It is now widely recognized that W-infinity algebras of both types are deeply linked with integrability of nonlinear systems. A number of integrable systems are now known to be related to w∞ algebras. Even within the context of field theory, one can pick out several important examples such as: the dispersionless KP hierarchy [2][3][4], the SU(∞) Toda field theory [5][6][7], its hierarchy (the dispersionless Toda hierarchy) [8][9], the selfdual vacuum Einstein equation (selfdual gravity) [10][11][12][13], etc. In these integrable systems, w∞ algebras are realized as the Poisson algebra Poisson(Σ) or the algebra sdiff(Σ) of area-preserving diffeomorphisms on a two dimensional surface Σ. Penrose’s twistor theory [14] provides a unified framework for understanding integrability of these systems. One may naturally ask if these integrable systems of w∞ type have any integrable deformation associated with a W∞ algebra. The dispersionless KP and Toda hierarchies do have such a W∞ analogue, i.e., the ordinary KP and Toda hierarchies, which are of course integrable. A W∞ analogue of selfdual gravity is recently proposed by Strachan [15] as a Moyal algebraic deformation of the selfdual vacuum Einstein equation. The problem of proving its integrability, however, remains obscure. In this paper, we consider this integrability problem by means of soliton theoretical techniques rather than of twistor theory. In soliton theory, the notion of “dressing operators” plays a central role. Our strategy is to construct dressing operators for Strachan’s deformation of selfdual gravity. We will then derive a “factorization relation” that connects those dressing operators with their “initial values” on a two dimensional subspace of space-time. This technique is borrowed from a similar approach to the KP hierarchy [16] and the Toda hierarchy [17], and