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

Gutt, Simone; Rawnsley, John

Lett. Math. Phys. 66, No. 1-2, 123-139 (2003).

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.

Reviewer: Marcela Popescu (Craiova)

MSC:

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

Keywords:

momentum map; natural star product; symplectic manifold

Citations:

Zbl 0465.53025; Zbl 0992.53065; Zbl 0867.58061