The KP Equation from Plebanski and $SU(\infty)$ Self-Dual Yang-Mills

Abstract

Starting from a self-dual $SU(\infty)$ 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 $SU(\infty)$ 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 $SU(\infty)$ 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 $D=2+2,~SU^*(\infty)$ Yang-Mills instantons. These $ϕ$ diffeomorphisms preserve a volume-three-form and are, precisely, the ones which provide the Plebanski-KP correspondence.