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. In particular, we show that if N is a Spin(7)-manifold, then M cannot have holonomy contained in G_2 unless the N is in fact a Calabi–Yau fourfold and M is the product of a Calabi–Yau threefold and an interval. By inverting this construction, we give examples of SU(4) holonomy metrics starting from torsion-free SU(3)-structures. 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 on the Bryant–Salamon Spin(7) metric on the spinor bundle of S^4 and on flat mathbb {R}^8.