Spin(7) Instantons and Hermitian Yang-Mills Connections for the Stenzel Metric

We use the large isometry group of the Stenzel asymptotically conical Calabi-Yau metric on $T^{\star}S^{4}$ to study the relationship between the Spin(7) instanton and Hermitian-Yang Mills (HYM) equations. We reduce both problems to tractable ODEs and look for invariant solutions. In the abelian case, we establish local equivalence and prove a global nonexistence result. We analyze the nonabelian equations with structure group SO(3) and construct the moduli space of invariant Spin(7) instantons in this setting. This includes an explicit one parameter family of irreducible Spin(7) instantons only one of which is HYM. We thus negatively resolve the question regarding the equivalence of the two gauge theoretic PDEs. The HYM connections play a role in the compactification of this moduli space, exhibiting a phenomenon that we aim to further look into in future work.