Moduli Spaces of Contact Instantons

We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric $5$-manifolds and initiate the study of their structure. In the $K$-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is given by the index of an operator elliptic transverse to the Reeb foliation. The moduli spaces are shown to be K\"ahler when the $5$-manifold $M$ is Sasakian and hyperK\"ahler when $M$ is transverse Calabi-Yau. We show how the transverse index can be computed in various cases, in particular we compute the index for the toric Sasaki-Einstein spaces $Y^{p,q}$.