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

Abstract

AbstractIf a Spin(7)-manifold $$N^8$$ N 8 admits a free $$S^1$$ S 1 action preserving the fundamental 4-form, then the quotient space $$M^7$$ M 7 is naturally endowed with a $$G_2$$ G 2 -structure. We derive equations relating the intrinsic torsion of the Spin(7)-structure to that of the $$G_2$$ G 2 -structure together with the additional data of a Higgs field and the curvature of the $$S^1$$ 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 $$G_2$$ 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 $$S^1$$ S 1 -quotient construction in detail on the Bryant–Salamon Spin(7) metric on the spinor bundle of $$S^4$$ S 4 and on flat $$\mathbb {R}^8$$ R 8 .