The Kp Equation from Plebanski and Su (∞) Self-dual Yang-mills

Starting from a self-dual SU(∞) Yang-Mills theory in (2 + 2) dimensions, the Plebanski second heavenly equation is obtained after a suitable dimensional reduction. The self-dual gravitational background is the cotangent space of the internal two-dimensional Riemannian surface required in the formulation of SU(∞) Yang-Mills theory. A subsequent dimensional reduction leads to the KP equation in (1 + 2) dimensions after the relationship from the Plebanski second heavenly function, Ω, to the KP function, u, is obtained. Also a complexified KP equation is found when a different dimensional reduction scheme is performed . Such relationship between Ω and u is based on the correspondence between the SL(2, R) self-duality conditions in (3 + 3) dimensions of Das, Khviengia, Sezgin (DKS) and the ones of SU(∞) in (2 + 2) dimensions . The generalization to the Supersymmetric KP equation should be straightforward by extending the construction of the bosonic case to the previous Super-Plebanski equation, found by us in [1], yielding self-dual supergravity backgrounds in terms of the light-cone chiral superfield, Θ, which is the supersymmetric analog of Ω. The most important consequence of this Plebanski-KP correspondence is that W gravity can be seen as the gauge theory of φ-diffeomorphisms in the space of dimensionally-reduced D = 2 + 2, SU∗(∞) Yang-Mills instantons. These φ diffeomorphisms preserve a volume-three-form and are, precisely, the ones which provide the Plebanski-KP correspondence.