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Variations on prequantization. (English) Zbl 1119.53060
Molitor-Braun, Carine (ed.) et al., Proceedings of the 4th conference on Poisson geometry, Luxembourg, June 7–11, 2004. Luxembourg: Université du Luxembourg (ISBN 2-87971-253-X/pbk). Travaux Mathématiques 16, 187-219 (2005).
Generalizations of prequantization are studied. First the prequantization procedure for symplectic manifolds is extended to Dirac manifolds. This includes presymplectic as well as Poisson manifolds. The paper starts with a concise review of Dirac and Dirac-Jacobi manifolds. Then the prequantization condition, the prequantization space and circle bundle and the prequantization representation on Dirac manifolds are systematically developed in both the U(1) and the line bundle approach. Contact manifolds are treated as an example. Finally, prequantization for U(1) actions with fixed points on a boundary is discussed. Remarks on cohomological prequantization, i.e. considering sheaves of local sections and noncommutative prequantization finish the paper.
For the entire collection see [Zbl 1085.53003].

MSC:
53D50 Geometric quantization
81S10 Geometry and quantization, symplectic methods
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