$\mathrm{G}_2$-instantons on the 7-sphere

Abstract

We study the deformation theory of $\mathrm{G}_2$-instantons on the 7-sphere, specifically those obtained from instantons on the 4-sphere via the quaternionic Hopf fibration. We find that the pullback of the standard ASD instanton lies in a smooth, complete, 15-dimensional family of $\mathrm{G}_2$-instantons. In general, the space of infinitesimal $\mathrm{G}_2$-instanton deformations on $S^7$ is identified with three copies of the space of ASD deformations on $S^4.$