Vierbein singularities in CP2

The compact open four-dimensional manifold $\text{C}$P², with Euclidean metric, has recently attracted attention as an example of a spacetime in which fields of half-integral spin cannot be defined in the absence of additional structure (such as an electromagnetic background). In this note we identify the specific topological anomaly responsible for this phenomenon: $\text{C}$P² contains a class of two-dimensional spheres—“complex lines”— in small neighborhoods of each of which the two transverse degrees of freedom are forced to “twist” in a characteristic way. It is shown in detail how the twists force vierbeins—tetrads of orthonormal vector fields that play a central role in the theory of spinors—to have singularities on every complex line. As an aid to visualization we construct an example of a vierbein with, loosely speaking, the smallest possible set of singularities: It is ill-defined at every point of some one complex line and smooth everywhere else. The behavior of such a vierbein near any one of its singular points is characterized explicitly. The structure of the minimally singular vierbein is used to illuminate the observation of Hawking and Pope that in the presence of an appropriate electromagnetic background, fields of any spin can exist on $\text{C}$P² as long as their electric charges are correctly quantized, but that the charge values available to half-integral-spin fields differ from those available to integral-spin fields.