# Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds

As is well-known (see Friedrich's book for example) every Kähler manifold is spin (or at least spin$^c$) and the Dirac is given (up to a twist) by $\partial + \partial^*$. What happens in the quaternionic-Kaehler and hyper-Kähler cases? Are they spin$^c$, does the Dirac admit a nice description? I'm particularly interested in the case of Wolf spaces (Quaternion-Kähler symmetric space).

EDIT: There is a hierarchy of groups $$Spin(n)\hookrightarrow Spin^c(n)=Spin(n)\cdot U(1)\hookrightarrow Spin^h(n)=Spin(n)\cdot Sp(1)\twoheadrightarrow SO(n)\;.$$ Here "$\cdot$" means "product divided by the diagonal $\mathbb Z/2$ action". If there is a principal bundle $P$ with structure group $Spin^x$ together with an equivariant map to the $SO(n)$ frame bundle of an oriented Riemannian manifold, then $P$ is called a spin$^x$ structure ($x$ being either void, or $c$, or $h$). Associated to $P$ is a real, complex, or quaternionic spinor bundle, respectively.
Every QK manifold comes with a natural spin$^h$ structure. For the Wolf spaces, these are automatically equivariant, see this answer. One of the Dirac operators in [op. cit.] acts on the corresponding quaternionic spinor bundles. Some Wolf spaces are spin, e.g. $\mathbb HP^k$, $G_2(\mathbb C^n)$ or $G_4(\mathbb R^n)$, the last two for $n$ even. Some are spin$^c$ but not spin, e.g. $G_2(\mathbb C^n)$ , and some are not even spin$^c$, e.g. $G_4(\mathbb R^n)$ for odd $n$, if I am not mistaken.
• A reference for the spin$^h$ structure on (almost) quaternionic manifolds is Lemma 3.9 of Elliptic Symbols by Bar. Mar 4, 2021 at 23:10
It is a result of Salamon that $8n$-dimensional quaternion-Kähler manifolds are spin.