W-Geometry from Fedosov's Deformation Quantization

Abstract

A geometric derivation of $W_{\infty }$ Gravity based on Fedosov's deformation quantization of symplectic manifolds is presented. To lowest order in Planck's constant it agrees with Hull's geometric formulation of classical nonchiral $W_{\infty }$ Gravity. The fundamental object is a $\mathcal{W}$-valued connection one form belonging to the exterior algebra of the Weyl algebra bundle associated with the symplectic manifold. The $\mathcal{W}$-valued analogs of the Self Dual Yang Mills equations, obtained from a zero curvature condition, naturally lead to the Moyal Plebanski equations, furnishing Moyal deformations of self dual gravitational backgrounds associated with the complexified cotangent space of a two dimensional Riemann surface. Deformation quantization of $W_{\infty }$ Gravity is retrieved upon the inclusion of all the $\hslash$ terms appearing in the Moyal bracket. Brief comments on Non Commutative Geometry and M(atrix)theory are made.

15 pages, revised Tex file. A discussion of the chiral approach to Self Dual Gravity is added to the previous Yang Mills one

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