Locally conformally symplectic bundles

Abstract

A locally conformally symplectic (LCS) form is an almost symplectic form $\omega$ such that a closed one-form $\theta$ exists with $d\omega=\theta\wedge\omega$. A fiber bundle with LCS fiber $(F, \omega,\theta)$ is called LCS if the transition maps are diffeomorphisms of $F$ preserving $\omega$ (and hence $\theta$). In this paper, we find conditions for the total space of an LCS fiber bundle to admit an LCS form which restricts to the LCS form of the fibers. This is done by using the coupling form introduced by Sternberg and Weinstein, \cite{gls}, in the symplectic case. The construction is related to an adapted Hamiltonian action called twisted Hamiltonian which we study in detail. Moreover, we give examples of such actions and discuss compatibility properties with respect to LCS reduction of LCS fiber bundles. We end with a glimpse towards the locally conformally K\"ahler case.