Formula omitted] geometry from Fedosov's deformation quantization

Abstract

A geometric derivation of W ∞ 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 non-chiral W ∞ gravity. The fundamental object is a W -valued connection one form belonging to the exterior algebra of the Weyl algebra bundle associated with the symplectic manifold. The 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 ∞ gravity is retrieved upon the inclusion of all the ħ terms appearing in the Moyal bracket. Brief comments on non commutative geometry and M(atrix) theory are made.