Complex $G_2$-manifolds and Seiberg-Witten Equations

We introduce the notion of complex $G_2$ manifold $M_{\mathbb C}$, and complexification of a $G_2$ manifold $M\subset M_{\mathbb C}$. As an application we show the following: If $(Y,s)$ is a closed oriented $3$-manifold with a $Spin^{c}$ structure, and $(Y,s)\subset (M, φ)$ is an imbedding as an associative submanifold of some $G_2$ manifold (such imbedding always exists), then the isotropic assoc…