Kirill Krasnov (Nottingham) will speak in the geometry seminar at 2pm on 21/03/2023 in the Salle des Profs (9th floor, building NO). Kirill’s title is “Lorentzian Cayley Form” and his abstract is below.
A suitably non-degenerate triple of 2-forms in 4D defines a metric. This works for all three possible signatures. However, one only gets the Riemannian and split signature metrics with real 2-forms, while complex 2-forms satisfying a certain reality condition give rise to Lorentzian metrics.
A Cayley form in 8D is a suitably non-degenerate real 4-form of a special algebraic type. It defines a metric in 8D, and this can be of either Riemannian or split signature. The 4D story with 2-forms can be reproduced from 8D by a variant of dimensional reduction (related to calibrations). However, this only reproduces the Riemannian and split versions of the 4D story. A natural question is whether exists some modification of the Cayley form that is capable of reproducing the Lorentzian version of the 4D geometry of triple of 2-forms.
The talk will provide a positive answer to this question. I will start by defining the notion of a complex Cayley form, which is a complex 4-form in 8D whose compatible metric is real. It will then be seen that a special complex Cayley form can be called Lorentzian. It is a complex 4-form in 8D that defines a metric of split signature and is calibrated by Lorentzian 4-dimensional subspaces. The dimensional reduction then leads to the sought Lorentzian version of the 4D story.
The 8D geometry that I describe relies on properties of (real and complex) spinors of Spin(8) and Spin(4,4).