Weyl manifold and quantized connection (Q1967134)

The subject is a geometric description of deformation quantization, an infinite dimensional Weyl algebra bundle over a symplectic manifold which is called a Weyl manifold. The relation between Weyl manifolds and deformation quantization is reviewed, and also a review is given of the Poincaré-Cartan class which gives the complete classification of Weyl manifolds. A quantized connection or twisted exterior derivative is discussed on a Weyl manifold, where the degree operator field plays an important role. It is shown that the quantized connection coincides with a Fedosov connection in terms of classical coordinates and then it is obtained that the Poincaré-Cartan class is equal to the deRham cohomology class of the curvature of the Fedosov connection.