W geometry from Fedosov's deformation quantization

A geometric derivation of W-infinity gravity based on Fedosov's deformation quantization of symplectic manifolds is presented. To lowest order in Plan ck's constant it agrees with Hull's geometric formulation of classical non- chiral W-infinity gravity. The fundamental object is a W-valued connection one form belonging to the exterior algebra of the Weyl algebra bundle assoc iated with the symplectic manifold. The W-valued analogs of the self-dual Y ang-Mills equations, obtained from a zero curvature condition, naturally le ad to the Moyal Plebanski equations, furnishing Moyal deformations of self- dual gravitational backgrounds associated with the complexified cotangent s pace of a two-dimensional Riemann surface. Deformation quantization of W-in finity, gravity is retrieved upon the inclusion of all the (h) over bar ter ms appearing in the Moyal bracket. Brief comments on non commutative geomet ry and M(atrix) theory are made. (C) 2000 Elsevier Science B.V. All rights reserved.