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 \( {\mathbbm{CP}}^2 \) , 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)max, where SO(3)max is maximal in SU(3). Intriguingly, the Wu manifold does not admit a spin structure or spinc structure, it does admit a spinh structure. We provide a physical interpretation of the spinh 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 spinh spinor harmonics from scalar harmonics. We provide a very explicit construction of all the scalar and spinh 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.
