Special metrics and Triality

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 K\"ahler 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.