Universality of Fedosov's construction for star products of Wick type on pseudo-Kähler manifolds

Abstract

In this paper we construct star products on a pseudo-Kähler manifold ( M, ω, I) using a modification of the Fedosov method based on a different fibrewise product similar to the Wick product on C n . Having fixed the used connection to be the pseudo-Kähler connection these star products shall depend on certain data given by a formal series of closed two-forms on M and a certain formal series of symmetric contravariant tensor fields on M. In a first step we show that this construction is rich enough to obtain star products of every equivalence class by computing Deligne's characteristic class of these products. Among these products we uniquely characterize the ones which have the additional property to be of Wick type which means that the bidifferential operators describing the star products only differentiate with respect to holomorphic directions in the first argument and with respect to anti-holomorphic directions in the second argument. These star products are in fact strongly related relvtar products with separation of variables introduced and studied by Karabegov. This characterization gives rise to special conditions on the data that enter the Fedosov procedure. Moreover, we compare our results that are based on an obviously coordinate independent construction to those of Karabegov that were obtained by local considerations and give an independent proof of the fact that star products of Wick type are in bijection to formal series of closed two-forms of type (1, 1) on M. Using this result we finally succeed in showing that the given Fedosov construction is universal in the sense that it yields all star products of Wick type on a pseudo-Kähler manifold. Due to this result we can make some interesting observations concerning these star products; we can show that all these star products are of Vey type and in addition we can uniquely characterize the ones that have the complex conjugation incorporated as an anti-automorphism.