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

Abstract

We use the highly symmetric Stenzel Calabi–Yau structure on $$T^{\star }S^{4}$$ T ⋆ S 4 as a testing ground for 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 is comprised of two 1-parameter families—one of them explicit—of irreducible Spin(7) instantons. Each carries a unique HYM connection. 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 removable singularity phenomenon that we aim to further examine in future work.

Examples of deformed G2-instantons/Donaldson–Thomas connections

“…In particular, this shows that deformed Spin(7)-instantons are not equivalent to dHYM connections with phase 1 when M has holonomy contained in SU(4) but is non-compact; this is rather different from the compact situation as shown by Kawai-Yamamoto in [23,Theorem 7.5], see also Theorem 2.4 (2) below. An analogous result was shown by Papoulias in [31] where the author showed that Spin(7)-instantons and traceless HYM connections do not coincide on the Calabi-Yau manifold T * S 4 ; this is again in contrast with the compact case as demonstrated by Lewis in [28,Theorem 3.1], see also [23,Remark 7.6]. This discrepancy is essentially due to the fact that the arguments in the compact cases in [23] and [28] both rely on certain energy estimates and these do not hold in the non-compact setting.…”