Deformation Quantization of Lagrangian Fiber Bundles

Details Deformation Quantization of Lagrangian Fiber Bundles Deformation Quantization of Lagrangian Fiber Bundles

1999-07-24

arxiv.org/abs/math/9907164v2

Conference Moshe Flato 1999, vol. 2, 269-288, Kluwer Acad. Publ. 2000

Mathematics

Quantum Algebra

24 pages, Latex2e, corrected typos

Scientific paper

Let (M, \om) be a symplectic manifold. A Lagrangian fiber bundle \pi : M -> B determines a completely integrable system on M. First integrals of this system are the pull-backs of functions on the base of the bundle. We show that for each Lagrangian fiber bundle \pi there exist star products on C^\infty(M)[[h]] which do not deform the pointwise multiplication on the subalgebra \pi^*(C^\infty (B)) [[h]]. The set of equivalence classes of such star products is in bijection with formal deformations of the symplectic structure \om for which \pi : M -> B remains Lagrangian taken modulo formal symplectomorphisms of M.

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