Complex $G_2$-manifolds and Seiberg-Witten Equations

TL;DR

This paper introduces complex $G_2$-manifolds and demonstrates that associative deformations of 3-manifolds within their complexifications are governed by Seiberg-Witten equations, linking geometric structures to gauge theory.

Contribution

It defines complex $G_2$-manifolds and establishes a novel connection between associative deformations and Seiberg-Witten equations in this setting.

Findings

Associative deformations are described by Seiberg-Witten equations.

Complex $G_2$-manifolds generalize real $G_2$-structures.

Existence of associative embeddings for 3-manifolds.

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 $Spin^{c}$ structure, and $(Y,s)\subset (M, \varphi)$ 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.