Gerbes, 2-gerbes and symplectic fibrations

Abstract

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