Fedosov manifold explained

In mathematics, a Fedosov manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (M, ω, ∇), where (M, ω) is a symplectic manifold (that is,

\omega is a symplectic form, a non-degenerate closed exterior 2-form, on a C ^{infty} -manifold M), and ∇ is a symplectic torsion-free connection on M. ^[1] (A connection ∇ is called compatible or symplectic if X ⋅ ω(Y,Z) = ω(∇_XY,Z) + ω(Y,∇_XZ) for all vector fields X,Y,Z ∈ Γ(TM). In other words, the symplectic form is parallel with respect to the connection, i.e., its covariant derivative vanishes.) Note that every symplectic manifold admits a symplectic torsion-free connection. Cover the manifold with Darboux charts and on each chart define a connection ∇ with Christoffel symbol \Gamma ^{i} _{jk} =0 . Then choose a partition of unity (subordinate to the cover) and glue the local connections together to a global connection which still preserves the symplectic form. The famous result of Boris Vasilievich Fedosov gives a canonical deformation quantization of a Fedosov manifold.^[2]

Examples

For example,

\R ^{2n} with the standard symplectic form dx _{i} \wedge dy _{i} has the symplectic connection given by the exterior derivative d. Hence, \left(\R ^{2n} , \omega, d\right) is a Fedosov manifold.

References

• 1305.2852 . Esrafilian . Ebrahim . Hamid Reza Salimi Moghaddam . Symplectic Connections Induced by the Chern Connection . 2013 . math.DG .

Notes and References

1. Gelfand . I. . Retakh . V. . Shubin . M. . dg-ga/9707024 . Fedosov Manifolds . Preprint . 1997 . 1997dg.ga.....7024G .
2. Fedosov. B. V.. A simple geometrical construction of deformation quantization. Journal of Differential Geometry. 40. 1994. 2. 213–238. 10.4310/jdg/1214455536. 1293654. free.