CMSA Geometry and Mathematical Physics Seminar: A tale of two Lie groups

The classical Lie group SO(4) is well-known to possess a very rich structure, relating in several ways to complex Euclidean spaces. This structure can be used to construct the classical twistor space Z over an oriented Riemannian 4-manifold M, which is a 6-dimensional almost Hermitian manifold. Special geometric properties of Z are then related to the curvature of M, an example of which is the celebrated Atiyah-Hitchin-Singer Theorem. The Lie group Spin(7) is a particular subgroup of SO(8) determined by a special 4-form. Inriguingly, Spin(7) has several properties relating to complex Euclidean spaces which are direct analogues of SO(4) properties, but sadly (or interestingly, depending on your point of view) not all of them. I will give a leisurely introduction to both groups in parallel, emphasizing the similarities and differences, and show how we can nevertheless at least partially succeed in constructing a “twistor space” over an 8-dimensional manifold equipped with a torsion-free Spin(7)-structure. (I will define what those are.) This is joint work with Michael Albanese, Lucía Martín-Merchán, and Aleksandar Milivojevic.