Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The geometry of three-forms in six and seven dimensions

Nigel Hitchin

TL;DR

The paper develops a diffeomorphism-invariant functional on the space of $3$-forms in dimensions $6$ and $7$, linking open $GL$-orbits to reductions of the structure group to $SL(3,\mathbf{C})$ and $G_2$. By restricting to closed $3$-forms in a fixed cohomology class and analyzing critical points, it shows that generic critical points yield integrable complex structures on $6$-manifolds (Calabi–Yau-type) and metrics with holonomy $G_2$ in $7$-dimensions, respectively. It then proves formal Morse–Bott nondegeneracy and, via a Banach-space implicit function theorem, constructs local moduli spaces that are open in $H^3(M,\mathbf{R})$, endowed with natural special (pseudo-)Kähler geometry. The results provide a direct variational route to deformation theory in these settings, clarifying the geometric structures arising from 3-forms and offering a framework aligned with mirror symmetry and Joyce–Tian–Todorov perspectives.

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.

The geometry of three-forms in six and seven dimensions

TL;DR

The paper develops a diffeomorphism-invariant functional on the space of -forms in dimensions and , linking open -orbits to reductions of the structure group to and . By restricting to closed -forms in a fixed cohomology class and analyzing critical points, it shows that generic critical points yield integrable complex structures on -manifolds (Calabi–Yau-type) and metrics with holonomy in -dimensions, respectively. It then proves formal Morse–Bott nondegeneracy and, via a Banach-space implicit function theorem, constructs local moduli spaces that are open in , endowed with natural special (pseudo-)Kähler geometry. The results provide a direct variational route to deformation theory in these settings, clarifying the geometric structures arising from 3-forms and offering a framework aligned with mirror symmetry and Joyce–Tian–Todorov perspectives.

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.

Paper Structure

This paper contains 26 sections, 21 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Linear algebra
  3. The complex case
  4. The real case
  5. Symplectic geometry
  6. The moment map
  7. The Hamiltonian function $\phi$
  8. The complex structure on $U$
  9. Special geometry of $U$
  10. Self-duality of $3$-forms
  11. Lorentzian self-duality
  12. Lagrangian aspects
  13. Spinor formulation
  14. An invariant functional
  15. Critical points
  16. ...and 11 more sections

Key Result

Proposition 1

For $\Omega\in \Lambda^3V^*$, $\lambda(\Omega)\ne 0$ if and only if $\Omega=\alpha+\beta$ where $\alpha,\beta$ are decomposable and $\alpha\wedge \beta\ne 0$. The $3$-forms $\alpha,\beta$ are unique up to ordering.

Theorems & Definitions (25)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Definition 3
  • Proposition 7
  • ...and 15 more