Gerbes, 2-gerbes and symplectic fibrations

Details Gerbes, 2-gerbes and symplectic fibrations Gerbes, 2-gerbes and symplectic fibrations

2005-04-13

arxiv.org/abs/math/0504274v1

Mathematics

Differential Geometry

42 pages, 21 references

Scientific paper

Let (F,u)\to P\to N be a symplectic fibration in math.SG/0503268 McDuff has defined a subgroup Ham^s(F,u) of the group of symplectic automorphisms of(F,u). She has shown that the cohomology class [u] of u can be extended to P if and only if the symplectic fibration has an Ham^s reduction. To show this result, she constructs a class who represents the obstruction to extend u. This class can be identified to a 3-class of P using a spectral sequence. The purpose of this paper is to define a 2-gerbe whose classifying cocycle is the class defined by McDuff. To define this 2-gerbe, we construct fundamental gerbes in Dirac geometry which represents the obstruction of [u] to be exact or integral. Using this gerbes we propose a quantization of symplectic manifolds

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