Vortices and a TQFT for Lefschetz fibrations on 4–manifolds

Abstract

Adapting a construction of D Salamon involving the $U\left(1\right)$ vortex equations, we explore the properties of a Floer theory for 3–manifolds that fiber over $S^1$ which exhibits several parallels with monopole Floer homology, and in all likelihood coincides with it. The theory fits into a restricted analogue of a TQFT in which the cobordisms are required to be equipped with Lefschetz fibrations, and has connections to the dynamics of surface symplectomorphisms.