Duality and fibrations on $G_2$ manifolds

Abstract

We argue that $G_2$ manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M-five-brane fibers as target space. Mirror symmetry and various string and M-theory dualities involving $G_2$ manifolds may be incorporated into this framework. To give some examples, we construct two non-compact manifolds with $G_2$ structures: one with a K3 fibration, and one with a torus fibration and a metric of $G_2$ holonomy. Kaluza-Klein reduction of the latter solution gives abelian BPS monopoles in 3 +1 dimensions.