$S^1$-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 R 8 .

Spin(7) metrics from Kähler Geometry

“…Note in particular that (2.1) implies that ϕ is a closed G 2 -structure. Moreover, from [14,Theorem 3.6 ] we also know that ϕ is also coclosed, hence torsion free, if and only if g Φ has holonomy contained in SU (4).…”

“…Note that even if N 8 is a hyperKähler manifold it is not generally the case that the quotient SU (3)-structure is torsion free. For instance, in [14] we considered the T 2quotient, generated by right and left multiplication by an imaginary quaternion, of (an open set in) R 8 ∼ = H 2 with the flat Spin(7)-structure and found that the quotient SU (3)-structure has all of π 1 , π 2 and σ 2 non-zero.…”

A circle quotient of a $G_2$ cone

“…cos 2θ)dφ 2 , lim v−>0 g 2 = |dτ | 2 + 1 cos 2θ)dφ 2 ,limits that are verified using Theorem 4.10. Although g 1 has little to do with G 2 holonomy, it is associated to a G 2 structure with closed 3-form that arises as the U (1) quotient of the flat Spin(7)-structure on R 8[16].The 4-dimensional subvarieties F ± of M are distinguished by their SO(3)-invariance. Neither can be J-holomorphic; this follows from Lemma 8.5.…”