Deformation quantization of symplectic fibrations. (English) Zbl 0992.53065

Kravchenko, Olga

Compos. Math. 123, No. 2, 131-165 (2000).

Quantization is a map from functions on a phase space \(M\) to operators on some Hilbert space. The product of operators induces a noncommutative product, called *-product, on the algebra \(C^\infty(M)[[h]]\) of formal power series in Planck’s constant \(h\). The *-algebra obtained this way is called a deformation quantization of \(C^\infty (M)\).
A symplectic fibration is a bundle of symplectic fibers over a symplectic base with a symplectic structure group. The main result of the paper describes the relation between the deformation quantization of the base and the fiber, and that of the total space. The deformation quantization involved is the one constructed by Feodosov using Koszul-type resolutions. The author generalizes Feodosov’s construction to the quantization with values in a bundle of algebras. She introduces the notion of an \(F\)-manifold consisting of a triple (manifold, deformation of symplectic form, connection), and then recalls a theorem of Feodosov stating that an \(F\)-manifold uniquely determines a *-product on the underlying manifold.
The main theorem refers to fibrations for which both the fiber and the base are \(F\)-manifolds. It shows that the quantization of the fiber yields a bundle of algebras over the given base, and this is a quantization of the base with values in the bundle of algebras. Finally, this defines a quantization of the total space of the fibration.
The paper is structured as follows: section 1 presents the main result with the subsequent definitions and some background material. Section 2 presents generalities on deformation quantization, such as the Weyl algebra of a vector space, symplectic connections, the definition of deformation quantization of a symplectic manifold, and introduces the notion of quantization with values in a bundle of algebras. The next section discusses symplectic forms on symplectic fibrations constructing a one-parameter family of symplectic forms on the total space. Section 4 defines the quantum moment map, which is a deformation with respect to \(h\) of the classical moment map. Section 5 contains the proof of the main result. The last section shows some examples of symplectic fibrations and their quantization, such as cotangent bundles to fiber bundles, and sphere bundles.

Reviewer: Razvan Gelca (Lubbock)

MSC:

53D55	Deformation quantization, star products
53D20	Momentum maps; symplectic reduction
81S10	Geometry and quantization, symplectic methods
32G08	Deformations of fiber bundles

Keywords:

deformation; quantization; symplectic fibration; star products; \(F\)-manifold