How to Calculate the Fedosov Star–product (exercices De Style)

This is an expository note on Fedosov’s construction of deformation quantization. Given a symplectic manifold and a connection on it, we show how to calculate the star-product step by step. We draw simple diagrams to solve the recursive equations for the Fedosov connection and for flat sections of the Weyl algebra bundle corresponding to functions. We also reflect on the differences of symplectic and Riemannian geometries. Key–words: deformation quantization, Weyl algebra, Fedosov quantization AMS classification (2000): 53D55 1. Deformation quantization of a symplectic manifold 1.1. Fedosov’s idea: Koszul–type resolution. We consider a deformation quantization of a symplectic manifold (M,ω0) as a deformation of an algebra of smooth functions on M in the direction of the Poisson bracket [1]. Definition 1.1. Deformation quantization of a symplectic manifold (M,ω0) is an associative algebra structure on A = C(M)[[t]] over C[[t]], called a ∗-product, such that for any a = a(x, t) = ∑∞ k=0 t ak(x) and b = b(x, t) = ∑∞ k=0 t bk(x), ak(x), bk(x) ∈ C (M) 1. The product ∗ is local, that is in the ∗-product a(x, t) ∗ b(x, t) = ∑∞ k=0 t ck(x), the coefficients ck(x) depend only on ai, bj and their derivatives ∂ ai, ∂ b with i+ j + |α|+ |β| ≤ k. 2. It is a formal deformation of the commutative algebra C(M) : c0(x) = a0(x)b0(x). 3. Let {·, ·} be the Poisson bracket of functions, given by a bivector field dual to the form ω0. There is a correspondence principle: [a, b] := i t (a ∗ b− b ∗ a) = {a0(x), b0(x)} + t r(a, b), where r(a, b) ∈ A. 4. There is a unit: a(x, t) ∗ 1 = 1 ∗ a(x, t) = a(x, t). Fedosov found a geometric way to perform the deformation quantization [4, 5] (also see [9] for a comprehensive exposition). The following idea lies behind Fedosov’s construction — a Koszul–type Presented in RENCONTRES MATHMATIQUES DE GLANON (Bourgogne, France) 1998 http://www.u-bourgogne.fr/glanon/ . 1