The geometry of three-forms in six and seven dimensions

TL;DR

This paper explores the algebraic properties of 3-forms in 6 and 7 dimensions, introducing a functional whose critical points correspond to special geometric structures like Calabi-Yau and G2 manifolds.

Contribution

It introduces a diffeomorphism-invariant functional on 3-forms that characterizes special geometric structures in six and seven dimensions.

Findings

Critical points define complex threefolds with trivial canonical bundle in 6D

Critical points define G2 holonomy manifolds in 7D

Provides a method to study moduli spaces of these structures

Abstract

We study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the functional to closed forms in a fixed cohomology class, we find that a critical point which is generic in a suitable sense defines in the 6-dimensional case a complex threefold with trivial canonical bundle and in 7 dimensions a Riemannian manifold with holonomy G2. This approach gives a direct method of finding a local moduli space, with its special geometry, for these structures.