Lorentzian Cayley form

Cayley four-form Φ on an eight-dimensional manifold M is a real differential form of a special algebraic type, which determines a Riemannian metric on M as well as a unit real Weyl spinor. It defines a Spin(7) structure on M, and this Spin(7) structure is integrable if and only if Φ is closed. We introduce the notion of a complex Cayley form. This is a one-parameter family of complex four-forms Φτ on M of a special algebraic type. Each Φτ determines a real Riemannian metric on M, as well as a complex unit Weyl spinor ψτ. The subgroup of GL(8,R) that stabilizes Φτ,τ≠0 is SU(4), and Φτ defines on M an SU(4) structure. We show that this SU(4) structure is integrable if and only if Φτ is closed. We carry out a similar construction for the split signature case. There are now two one-parameter families of complex Cayley forms. A complex Cayley form of one type defines an SU(2,2) structure, a form of the other type defines an SL(4,R) structure on M. As in the Riemannian case, these structures are integrable if and only if the corresponding complex Cayley forms are closed. Our central observation is that there exists a special member of the second one-parameter family of complex Cayley forms, which we call the Lorentzian Cayley form. This four-form has the property that it calibrates two four-dimensional subspaces H,H⊥ that have the property that the induced metric on H,H⊥ is Lorentzian. In particular, in a basis adapted to such a calibration, the Lorentzian Cayley form is built from the complex self-dual two-forms for H,H⊥. We explain how these observations solve a certain puzzle that existed in the context of four-dimensional Lorentzian geometry.