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We introduce the notion of complex $G_2$ manifold $M_{\mathbb C}$, andcomplexification of a $G_2$ manifold $M\subset M_{\mathbb C}$. As anapplication we show the following: If $(Y,s)$ is a closed oriented $3$-manifoldwith a $Spin^{c}$ structure, and $(Y,s)\subset (M, \varphi)$ is an imbedding asan associative submanifold of some $G_2$ manifold (such imbedding alwaysexists), then the isotropic associative deformations of $Y$ in the complexified$G_2$ manifold $M_{\mathbb C}$ is given by Seiberg-Witten equations.

Complex $G_2$-manifolds and Seiberg-Witten Equations