W Gravity, $N=2$ Strings and $2+2S{U}^{\ast }(\infty )$

Abstract

We conjecture that $W$ gravity can be interpreted as the gauge theory of $\varphi$ diffeomorphisms in the space of dimensionally-reduced $D=2+2$ $S{U}^{\ast }(\infty )$ Yang-Mills instantons. These $\varphi$ diffeomorphisms preserve a volume-three form and are those which furnish the correspondence between the dimensionally-reduced Plebanski equation and the KP equation in $(1+2)$ dimensions. A supersymmetric extension furnishes super-$W$ gravity. The Super-Plebanski equation generates self-dual complexified super gravitational backgrounds (SDSG) in terms of the super-Plebanski second heavenly form. Since the latter equation yields $N=1D=4SDSG$ complexified backgrounds associated with the complexified-cotangent space of the Riemannian surface, $(T^{\ast }\Sigma {)}^c$, required in the formulation of $S{U}^{\ast }(\infty )$ complexified Self-Dual Yang-Mills theory, (SDYM ); it naturally follows that the recently constructed $D=2+2N=4$ SDSYM theory- as the consistent background of the open $N=2$ superstring- can be embedded into the $N=1S{U}^{\ast }(\infty )$ complexified Self-Dual-Super-Yang-Mills (SDSYM) in $D=3+3$ dimensions. This is achieved after using a generalization of self-duality for $D>4$. We finally comment on the the plausible relationship between the geometry of $N=2$ strings and the moduli of $S{U}^{\ast }(\infty )$ complexified SDSYM in $3+3$ dimensions.

10 pages

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• W-Geometry from Fedosov's Deformation Quantization
• Infinite Conformal Algebras in Supersymmetric Theories on Four Manifolds
• The Generalized Moyal Nahm and Continuous Moyal Toda Equations
• A Moyal Quantization of the Continuous Toda Field
• Large Nonlinear $W_{\infty }$ Algebras from Nonlinear Integrable Deformations of Self Dual Gravity
• N=2 String Geometry and the Heavenly Equations