TITLE: Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds
QUESTION [7 upvotes]: 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).

REPLY [2 votes]: It is a result of Salamon that $8n$-dimensional quaternion-Kähler manifolds are spin.