  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.
