Local ν-Euler Derivations and Deligne's¶Characteristic Class of Fedosov Star Products¶and Star Products of Special Type

Abstract

In this paper we explicitly construct local ν-Euler derivations $\mathsf E_\alpha = \nu \partial_\nu + \Lie{\xi_\alpha} + \mathsf D_\alpha$ , where the ξα are local, conformally symplectic vector fields and the $\mathsf D_\alpha$ are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold (M,ω) by means of which we are able to compute Deligne's characteristic class of these star products. We show that this class is given by $\frac{1}{\nu}[\omega]+\frac{1}{\nu} [\Omega]$ , where $\Omega \in \nu Z^2_{{\rm dR}}(M)[[\nu]]$ is a formal series of closed two-forms on M the cohomology class of which coincides with the one introduced by Fedosov to classify his star products. Moreover, we consider star products that have additional algebraic structures and compute the effect of these structures on the corresponding characteristic classes of these star products. Specifying the constituents of Fedosov's construction we obtain star products with these special properties. Finally, we investigate equivalence transformations between such special star products and prove existence of equivalence transformations being compatible with the considered algebraic structures. Dedicated to the memory of Moshe Flato