A functional for Spin(7) forms

TL;DR

This paper characterizes conformal Spin(7) structures on 8-manifolds as solutions to a quadratic algebraic equation and constructs a functional whose critical points correspond to these structures, linking geometry and physics.

Contribution

It introduces a new algebraic characterization of conformal Spin(7) forms and constructs a functional with critical points exactly at these structures, connecting to Einstein-Hilbert action.

Findings

Characterization of conformal Spin(7) forms via quadratic algebraic equations

Construction of a functional with Spin(7) structures as critical points

Link between Spin(7) structures and conformally Ricci-flat geometries

Abstract

We characterize the set of all conformal Spin(7) forms on an oriented and spin Riemannian eight-manifold $(M,g)$ as solutions to a homogeneous algebraic equation of degree two for the self-dual four-forms of $(M,g)$. When $M$ is compact, we use this result to construct a functional whose self-dual critical set is precisely the set of all Spin(7) structures on $M$. Furthermore, the natural coupling of this potential to the Einstein-Hilbert action gives a functional whose self-dual critical points are conformally Ricci-flat Spin(7) structures. Our proof relies on the computation of the square of an irreducible and chiral real spinor as a section of a bundle of real algebraic varieties sitting inside the K\"ahler-Atiyah bundle of $(M,g)$.