Classification of symplectic non-Hamiltonian circle actions on 4-manifolds

We classify symplectic non-Hamiltonian circle actions on compact connected symplectic 4-manifolds, up to equivariant symplectomorphisms. Namely, we define a set of invariants, show that the set is complete, and determine which values are attainable by constructing a space for each valid choice. We work under the assumption that the group of periods of the one-form $\iota_X \omega$ is discrete, which allows us to define a circle-valued Hamiltonian for the action, and apply tools from Karshon-Tolman's work on the classification of complexity one spaces. This assumption is always satisfied if the symplectic form is rational, or if the quotient space has first Betti number one.