Geometry of Spin(7) structures in eight dimensions

Project description

Over the past 50 years, mathematicians have developed numerous examples where a pair (metric, spinor) on a manifold M is encoded into a suitable collection of differential forms on M. In four dimensions, this encoding translates such a pair into a triple of 2-forms that satisfy specific algebraic relations. Remarkably, the Einstein condition for the metric can be expressed through a simple set of differential equations in terms of this triple.

A similar encoding exists in eight dimensions, where the pair (metric, spinor) is represented by a single 4-form, again constrained by a set of algebraic relations. The appropriate language for this geometric setup is that of G-structures. A triple of 2-forms in four dimensions encodes an SU(2)-structure, a 4-form of a special algebraic type in eight dimensions encodes a Spin(7)-structure.

The goal of this project is to construct and study gravity theories in dimension eight that can be formulated using a 4-form rather than a metric as the main dynamical object.