1993/08/20 by Carlos Castro, Castro, Carlos
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9308101
10 pages, IAEC 7-93R
arxiv created 1993/08/20 · arxiv updated 2009/11/30
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.