Seiberg-Witten invariants on manifolds with Riemannian foliations of codimension 4

We define Seiberg-Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K-contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and we highlight that the invariants may be used to distinguish different foliations on diffeomorphic manifolds.