Maslov $S^{1}$ Bundles and Maslov Data

Abstract

We define Maslov $S^1$ bundles over a symplectic manifold $(M,ω)$. These are the determinant bundle $Γ_J$ of the unitary frame bundle defined by an almost complex structure compatible with $ω$, and the bundle $Γ_J^2 = Γ_J \big/ \{\pm1\}$. We analyze the properties of the Maslov $S^1$ bundles $Γ_J$ and $Γ_J^2$, 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 $Γ_J$ and on $Γ_J^2$. We show that when $M$ is a homogeneous $G$-space and the first real Chern class $c_Γ$ is nonvanishing, $Γ_J$ and $Γ_J^2$ are also homogeneous $G$-spaces. Moreover, we give an alternative proof of the fact that when $[ω]=r\,c_Γ$ for some real number $r$, then the symplectic $G$ action on $(M,ω)$ is Hamiltonian. When the Maslov $S^1$ bundle $Γ_J^2$ is trivial, then an index generalizing the Maslov index can be defined. This is no longer true if $Γ_J^2$ is not trivial. However, if $G=S^1$ 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 $Γ_J^2$ is not trivial, and we associate the Maslov data at fixed points of the $G=S^1$ action to their resonance type. Finally, we consider three applications motivated by the study of integrable Hamiltonian systems. First, we discuss conditions under which an $S^1$ symmetry of a two degrees of freedom integrable Hamiltonian system can be extended to a $\mathbb T^2$ symmetry. Second, we show that the Maslov $S^1$ bundles over Lagrangian pinched tori are trivial. Third, we consider $S^2 \times S^2$ as a symplectic manifold with an $S^1$ action corresponding to simultaneous rotations of the two spheres, and we compute the corresponding Maslov data.