Local \(\nu\)-Euler derivations and Deligne’s characteristic class of Fedosov star products and star products of special type. (English) Zbl 1035.53124

The author gives an explicit construction of local \(\nu\)-Euler derivations \(E_\alpha= \nu\partial_\nu+{\mathcal L}_{\xi_\alpha}+ D_\alpha\), where the \(\xi_\alpha\) are local, conformally symplectic vector fields and the \(D_\alpha\) are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold \((M, \omega)\) by means of which the author computes Deligne’s characteristic class of these star products. As an application of the properties of Deligne’s characteristic class and its relation to Fedosov’s Weyl curvature the author studies star products of special type that satisfy special algebraic identities with respect to complex conjugation and the mapping \(\nu\to -\nu\) changing the sign of the formal parameter and computes the influence on the corresponding characteristic classes. It is proved that there are always Fedosov star products satisfying these special algebraic identities and that the corresponding characteristic class is given by \({1\over \nu}[\omega]+ {1\over\nu}[\Omega]\), where \(\Omega\in \nu Z^2_{dR}(M)[[\nu]]\) is a formal series of closed two-forms on \(M\) the cohomology class of which coincides with the one introduced by B. Fedosov to classify his star products [cf. his book “Deformation Quantization and Index Theory”. Berlin: Akademie Verlag (1996; Zbl 0867.58061) and the paper in J. Differ. Geom. 40, 213–238 (1994; Zbl 0812.53034)]. Considering equivalent star products satisfying the same algebraic identities with respect to the mappings mentioned it is also proved that there are always equivalence transformations between these star products commuting with these mappings.
The notions of Deligne’s intrinsic derivation-related class and Deligne’s characteristic class and the realitions between them, used by the author, are those due to P. Deligne [Sel. Math., New Ser. 1, 667–697 (1995; Zbl 0852.58033)].
Reviewer: Ioan Pop (Iaşi)


53D55 Deformation quantization, star products
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