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We conjecture that $W$ gravity can be interpreted as the gauge theory of$\phi$ diffeomorphisms in the space of dimensionally-reduced $D=2+2$$SU^*(\infty)$ Yang-Mills instantons. These $\phi$ diffeomorphisms preserve avolume-three form and are those which furnish the correspondence between thedimensionally-reduced Plebanski equation and the KP equation in $(1+2)$dimensions. A supersymmetric extension furnishes super-$W$ gravity. TheSuper-Plebanski equation generates self-dual complexified super gravitationalbackgrounds (SDSG) in terms of the super-Plebanski second heavenly form. Sincethe latter equation yields $N=1~D=4~SDSG$ complexified backgrounds associatedwith the complexified-cotangent space of the Riemannian surface,$(T^*\Sigma)^c$, required in the formulation of $SU^*(\infty)$ complexifiedSelf-Dual Yang-Mills theory, (SDYM ); it naturally follows that the recentlyconstructed $D=2+2~N=4$ SDSYM theory- as the consistent background of the open$N=2$ superstring- can be embedded into the $N=1~SU^*(\infty)$ complexifiedSelf-Dual-Super-Yang-Mills (SDSYM) in $D=3+3$ dimensions. This is achievedafter using a generalization of self-duality for $D>4$. We finally comment onthe the plausible relationship between the geometry of $N=2$ strings and themoduli of $SU^*(\infty)$ complexified SDSYM in $3+3$ dimensions.Comment: 10 page

W gravity, N=2 strings, and 2+2 SU*(∞) Yang–Mills instantons