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

TL;DR

This paper extends Seiberg-Witten invariants to manifolds with Riemannian foliations of codimension 4, establishing their properties and applications in distinguishing foliations.

Contribution

It introduces Seiberg-Witten equations on foliated manifolds and proves compactness, vanishing, and non-vanishing results for the associated invariants.

Findings

Compactness of the moduli space under certain conditions

Vanishing and non-vanishing of invariants

Invariants distinguish different foliations on the same manifold

Abstract

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.