Strong $G_2$-structures with torsion: a survey and new examples

Abstract

This article has two main parts. The first consists of a brief survey of $G_2$-structures with torsion, namely $G_2$-structures admitting a $G_2$-connection with totally skew-symmetric torsion. We focus primarily on the strong condition that the associated torsion $3$-form is closed. We review their basic properties and their connections with the heterotic $G_2$ system, generalized geometry, and six-dimensional geometry. In the second part, we construct two complete cohomogeneity-one strong $G_2T$-structures on $\mathbb{R}\times S^3\times S^3$ with non-zero closed and co-closed characteristic torsion, and non-vanishing characteristic Ricci tensor. Both have an asymptotically cylindrical end; at the other end, the first is asymptotically conical (AC), with a torsion-free $G_2$ cone as its asymptotic model, while the second is asymptotically locally conical (ALC), with a constant-length circle bundle over a Calabi--Yau cone as its asymptotic model. We investigate the induced $SU(3)$-structures on their circle quotients and show that they are globally conformally almost Kähler and globally conformally Kähler, respectively. To the best of our knowledge, these provide the first complete strong $G_2T$-structures whose underlying metrics are not locally Riemannian products.