Seiberg-Witten type equations on compact symplectic 6-manifolds

TL;DR

This paper introduces a higher-dimensional analogue of the Seiberg-Witten equations on compact symplectic 6-manifolds, defining a new invariant via gauge theory and establishing conditions for moduli space compactness.

Contribution

It formulates a 6-dimensional Seiberg-Witten type equation, constructs an associated invariant, and proves compactness of the moduli space on compact Kähler threefolds.

Findings

Defined a 6D Seiberg-Witten invariant using virtual neighbourhoods.

Proved moduli space compactness for Kähler threefolds.

Computed the invariant in specific cases.

Abstract

In this article, we consider a gauge-theoretic equation on compact symplectic 6-manifolds, which forms an elliptic system after gauge fixing. This can be thought of as a higher-dimensional analogue of the Seiberg-Witten equation. By using the virtual neighbourhood method by Ruan, we define an integer-valued invariant, a 6-dimensional Seiberg-Witten invariant, from the moduli space of solutions to the equations, assuming that the moduli space is compact; and it has no reducible solutions. We prove that the moduli spaces are compact if the underlying manifold is a compact Kahler threefold. We then compute the integers in some cases.