$\text{Spin}^h$ Structure, Scalar and Charged Spinor Eigenfunctions on the $SU(3)/SO(3)$ Wu Manifold
Generalised spin structures are necessary for placing fermions on manifolds that do not admit a standard spin structure. This is especially relevant in a dimensional reduction on such a manifold, which can then be compensated by using fermions that are appropriately charged under some Maxwell or Yang-Mills field defined on the internal manifold. A well known example in the physics literature is $\mathbb{CP}^2$, which has four real dimensions and is the coset $SU(3)/U(2)$. In this paper we focus on a five-dimensional coset space, namely the Wu manifold $SU(3)/SO(3)_{\rm max}$, where $SO(3)_{\rm max}$ is maximal in $SU(3)$. Intriguingly, the Wu manifold does not admit a spin structure or spin$^c$ structure, it does admit a spin$^h$ structure. We provide a physical interpretation of the spin$^h$ structure by considering spinors that are coupled to an $SO(3)$ Yang-Mills field defined on the Wu manifold, but which carry half-integer "isospin," thereby canceling the minus sign in the holonomy for uncharged spinors that provides the original obstruction to an ordinary spin structure. We also construct a gauge-covariantly constant spinor in the Wu manifold, and we show how this can be employed in order to construct spin$^h$ spinor harmonics from scalar harmonics. We provide a very explicit construction of all the scalar and spin$^h$ harmonics. In a follow-up paper, we shall employ the results we obtain here in order to discuss dimensional reductions and consistent reductions on the Wu manifold.