Natural star products on symplectic manifolds and quantum moment maps. (English) Zbl 1064.53061

The natural class of star products defined in the paper includes Vey star product as studied by A. Lichnerowicz in [Ann. Inst. Fourier 32, No. 1, 157–209 (1982; Zbl 0465.53025)], but also not Vey star products (as, for example, cotangent bundles or Kähler manifolds). In the symplectic context, these classes are parametrized in a covariant manner under diffeomorphisms in terms of a connection, a formal series of closed \(2 \)-forms and a formal series of symmetric tensors. Some invariance properties and necessary and sufficient conditions for the star products to have a quantum momentum map are also given. The authors also prove that O. Kravchenko’s sufficient condition [Compos. Math. 123, No. 2, 131–165 (2000; Zbl 0992.53065)] for a momentum map for a B. Fedosov star product [Deformation quantization and index theory. Mathematical Topics. 9. Berlin: Akademie Verlag (1996; Zbl 0867.58061)] is also necessary.


53D55 Deformation quantization, star products
53D20 Momentum maps; symplectic reduction
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