A New Construction for Non-Compact $G_2$ Manifolds

Constructing $G_2$ manifolds is a notoriously difficult task. Taking inspiration from physics, one can try to reverse engineer new $G_2$ manifolds by geometrising, via M-theory, the flux torus construction of 4d $\mathcal{N}=1$ quiver gauge theories. The resulting $G_2$ manifolds are described by highly non-trivial fibrations of an elliptically fibered Calabi-Yau threefold over a circle. In this talk I will describe the construction of a new infinite family of non-compact $G_2$ manifolds arising from flux torus compactifications of the rank 1 E-string theory. This talk will focus on the geometry and not the physics of this construction. Based on arXiv 2608.21238.