Special metrics and Triality

Abstract

We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator $\Db$ on $\Delta \otimes {\Lambda }^1$. We establish various obstructions to the existence of topological reductions to PSU(3). For compact manifolds, we also give sufficient conditions for topological PSU(3)-structures that can be lifted to topological SU(3)-structures. We also construct the first known compact example of an integrable non-symmetric PSU(3)-structure. In the same vein, we give a new Riemannian characterisation for topological quaternionic Kähler structures which are defined by an $Sp(1)\cdot Sp(2)$-invariant self-dual 4-form. Again, we show that this form is closed if and only if the corresponding spinor-valued 1-form is harmonic for $\Db$ and that these equivalent conditions produce constraints on the Ricci tensor.

41 pages. v2: typos corrected, presentation improved v3: final version

References in corpus (3)

• Special metric structures and closed forms
• Irreducible SO(3) geometry in dimension five
• Distinguished dimensions for special Riemannian geometries

Cited by in corpus (5)

• Closed forms and multi-moment maps
• The kernel of the Rarita-Schwinger operator on Riemannian spin manifolds
• Harmonic structures and intrinsic torsion
• Sp(3) structures on 14-dimensional manifolds
• Embedding into manifolds with torsion