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 $\mathsf E_α= ν\partial_ν+ \Lie{ξ_α} + \mathsf D_α$, where the $ξ_α$ are local, conformally symplectic vector fields and the $\mathsf D_α$ 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 $1/ν[ω]+1/ν[Ω]$ where $Ω= \sum_{i=1}^\infty ν^i Ω_i$ 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, our result implies that the normalisation condition used by Fedosov does not have any effect on the isomorphy class of the resulting star product. Finally 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.