A note on the index bundle over the moduli space of monopoles

Donaldson has shown that the moduli space of monopoles $M_k$ is diffeomorphic to the space ${\text{Rat}}_k$ of based rational maps from the two-sphere to itself. We use this diffeomorphism to give an explicit description of the bundle on ${\text{Rat}}_k$ obtained by pushing out the index bundle from $M_k$. This gives an alternative and more explicit proof of some earlier results of Cohen and Jones.