W-Geometry from Fedosov's Deformation Quantization

TL;DR

This paper derives $W_$ Gravity using Fedosov's deformation quantization, connecting classical and quantum geometries, and introduces Moyal deformations of self-dual gravitational backgrounds.

Contribution

It provides a geometric derivation of $W_$ Gravity from Fedosov's quantization, linking classical formulations with quantum deformations via Moyal brackets.

Findings

Agreement with Hull's classical $W_$ Gravity at lowest order

Derivation of Moyal Plebanski equations from zero curvature conditions

Connection between deformation quantization and noncommutative geometry

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 ${\cal W}$-valued connection one form belonging to the exterior algebra of the Weyl algebra bundle associated with the symplectic manifold. The ${\cal 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 $\hbar$ terms appearing in the Moyal bracket. Brief comments on Non Commutative Geometry and M(atrix)theory are made.