Maslov S 1 bundles and Maslov data

We define Maslov S1 bundles over a symplectic manifold (M, ω). These are the determinant bundle ΓJ of the unitary frame bundle defined by an almost complex structure compatible with ω, and the bundle ΓJ2=ΓJ/{±1}. We analyze the properties of the Maslov S1 bundles ΓJ and ΓJ2, focusing on the interplay between their geometry and the dynamics of a symplectic action of a compact Lie group G on M which induces lifted G actions on ΓJ and on ΓJ2. We show that when M is a homogeneous G-space and the first real Chern class cΓ is nonvanishing, ΓJ and ΓJ2 are also homogeneous G-spaces. Moreover, we give an alternative proof of the fact that when [ω] = r cΓ for some real number r, then the symplectic G action on (M, ω) is Hamiltonian. When the Maslov S1 bundle ΓJ2 is trivial, then an index generalizing the Maslov index can be defined. This is no longer true if ΓJ2 is not trivial. However, if G = S1 acts symplectically on (M, ω) we define a quantity that we call Maslov data which serves as a non-integrable version of the notion of Maslov index in the case where ΓJ2 is not trivial, and we associate the Maslov data at fixed points of the G = S1 action to their resonance type. Finally, we consider applications of Maslov bundles motivated by the study of integrable Hamiltonian systems.