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

Abstract

We define Maslov $S^1$ bundles over a symplectic manifold $(M,\omega)$. These are the determinant bundle $\Gamma_J$ of the unitary frame bundle defined by an almost complex structure compatible with $\omega$, and the bundle $\Gamma_J^2 = \Gamma_J \big/ \{\pm1\}$. We analyze the properties of the Maslov $S^1$ bundles $\Gamma_J$ and $\Gamma_J^2$, 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 $\Gamma_J$ and on $\Gamma_J^2$. We show that when $M$ is a homogeneous $G$-space and the first real Chern class $c_\Gamma$ is nonvanishing, $\Gamma_J$ and $\Gamma_J^2$ are also homogeneous $G$-spaces. Moreover, we give an alternative proof of the fact that when $[\omega]=r\,c_{\Gamma}$ for some real number $r$, then the symplectic $G$ action on $(M,\omega)$ is Hamiltonian. When the Maslov $S^1$ bundle $\Gamma_J^2$ is trivial, then an index generalizing the Maslov index can be defined. This is no longer true if $\Gamma_J^2$ is not trivial. However, if $G=S^1$ acts symplectically on $(M,\omega)$ 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 $\Gamma_J^2$ 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.