$S^1$-quotient of $Spin(7)$-structures

Abstract: If a $Spin(7)$ manifold $N^8$ admits a free $S^1$ action preserving the fundamental $4$-form then the quotient space $M^7$ is naturally endowed with a $G_2$-structure. We derive equations relating the intrinsic torsion of the $Spin(7)$-structure to that of the $G_2$-structure together with the additional data of a Higgs field and the curvature of the $S^1$-bundle; this can be interpreted as a Gibbons-Hawking-type ansatz for $Spin(7)$-structures. We focus on the three $Spin(7)$ torsion classes: torsion-free, locally conformally parallel and balanced. In particular we show that if $N$ is a $Spin(7)$ manifold then $M$ cannot have holonomy contained in $G_2$ unless $N$ is in fact a Calabi-Yau $4$-fold and $M$ is the product of a Calabi-Yau $3$-fold and an interval. We also derive a new formula for the Ricci curvature of $Spin(7)$-structures in terms of the torsion forms. We then describe this $S^1$-quotient construction in detail for the Bryant-Salamon $Spin(7)$ metric on the spinor bundle of $S^4$ and for the flat metric on $\mathbb{R}^8$.