Let $M$ be a quaternionic Kähler manifold, by which I understand a Riemannian manifold for which the holonomy group of its Levi-Civita connection is a subgroup of $Sp(1)Sp(n)$. Its complexified tangent bundle splits locally as $T_{\mathbb{C}}M = E \otimes H$, where $E$ and $H$ are locally-defined complex vector bundles of ranks $2n$ and $2$ associated to standard representations of $Sp(n)$ and $Sp(1)$. In four dimensions, where the isomorphism $Sp(1)Sp(1) \cong SO(4)$ renders the holonomy group characterization trivial, one takes instead $M$ to be Einstein and self-dual, and $E$ and $H$ to be the spinor bundles $S_+$ and $S_-$. This splitting is the cornerstone of Salamon’s so-called $E-H$ formalism.

My question is as follows: in four dimensions, totally symmetric sections of $H^{\otimes k}$ are usually referred to as spinors. Is that still an appropriate thing to call them in higher dimensions? Can one think of such sections as spinors in some sense when $n>1$?

Same question if the manifold is hyperkähler, that is, if the holonomy group is simply $Sp(n)$.


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