Wick type deformation quantization of Fedosov manifolds Original Research Article

A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The geometry of the symplectic manifolds admitting the symbol construction is explored and a certain analogue of the Newlander–Nirenberg theorem is presented. The 2-form is explicitly identified which cohomological class coincides with the Fedosov class of the Wick-type star-product. For the Kähler manifolds this class is shown to be proportional to the first Chern class of a complex manifold. We also show that the symbol construction admits canonical superextension, which can be thought of as the Wick-type deformation of the exterior algebra of differential forms on the base (even) manifold. Possible applications of the deformed superalgebra to the noncommutative field theory and strings are discussed.