Locally conformally symplectic bundles

TL;DR

This paper investigates conditions under which the total space of a locally conformally symplectic (LCS) fiber bundle admits an LCS form compatible with the fibers, extending symplectic coupling techniques to the LCS setting.

Contribution

It introduces conditions for LCS fiber bundles to admit compatible LCS forms on the total space, utilizing a coupling form approach and analyzing twisted Hamiltonian actions.

Findings

Derived conditions for total space LCS forms

Studied twisted Hamiltonian actions in LCS context

Explored compatibility with LCS reduction

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.