Triality rotation of connections and Dirac operators on SU(3)

We show how triality, the symmetry permuting the three 8-dimensional representations of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(8)$, can also appear as a symmetry of geometric objects such as connections and Dirac operators. Take any Riemannian metric $g$ on the manifold $M=\mathrm{S}\mathrm{U}(3)$ and let $\mathrm{\nabla }$ be a metric connection on $TM$. Using an explicit model of Clifford multiplication by deformed Okubo products, we combine spinorial lifts with isometries $TM\simeq S^{+}\simeq S^{-}$ to show that $\mathrm{\nabla }$ can be rotated into two new metric connections on $TM$. So there is a canonical ${\mathbb{Z}}_3$-orbit of metric connections determined by $(g,\mathrm{\nabla })$, with no other choices. The three connections coincide when $g$ is bi-invariant and $\mathrm{\nabla }$ belongs to the Cartan-Schouten line, but in general they are different. The rotations of $\mathrm{\nabla }$ preserve the norm of the curvature, so they define a new ${\mathbb{Z}}_3$-symmetry of the moduli space of Yang-Mills connections on $TM$. They also preserve the Chern-Simons 3-form and adjoint Yang-Mills-Higgs functionals. After choosing an orientation on $M$, the rotated connections define three self-adjoint, Dirac-type operators on the spinor bundle. We prove that these operators are natural, in the sense that they commute with the Kosmann derivatives of spinors along all invariant Killing vector fields that preserve $\mathrm{\nabla }$. We also show that they are not isospectral, in general. This is verified by computing the simplest eigenvalue for the Levi-Civita connection and observing that it splits among the three operators as the bi-invariant metric on $M$ is deformed to one with isometry group $(\mathrm{S}\mathrm{U}(3)\times \mathrm{S}\mathrm{U}(2)\times \mathrm{U}(1))\mathrm{/}{\mathbb{Z}}_6$. Finally, we outline how the constructions in the paper may extend to general 8-manifolds with a $\mathrm{P}\mathrm{S}\mathrm{U}(3)$-structure.