In this article, we consider a gauge-theoretic equation on compact symplectic6-manifolds, which forms an elliptic system after gauge fixing. This can bethought of as a higher-dimensional analogue of the Seiberg-Witten equation. Byusing the virtual neighbourhood method by Ruan, we define an integer-valuedinvariant, a 6-dimensional Seiberg-Witten invariant, from the moduli space ofsolutions to the equations, assuming that the moduli space is compact; and ithas no reducible solutions. We prove that the moduli spaces are compact if theunderlying manifold is a compact Kahler threefold. We then compute the integersin some cases.
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