Complex $G_2$-manifolds and Seiberg-Witten Equations

Abstract

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 $Spi{n}^c$ structure, and $(Y,s)\subset (M,\phi )$ is an imbedding as an associative submanifold of some $G_2$ manifold (such imbedding always exists), then the isotropic associative deformations of $Y$ in the complexified $G_2$ manifold $M_{\mathbb{C}}$ is given by Seiberg-Witten equations.

27 pages

References in corpus (2)

• Complex ${\text{G}}_2$ and Associative Grassmannian
• Calibrated associative and Cayley embeddings