Spin(7)-instantons on Joyce's first examples of compact Spin(7)-manifolds
Mateo Galdeano, Daniel Platt, Yuuji Tanaka, Luya Wang
TL;DR
This work constructs Spin(7)-instantons on Joyce’s compact Spin(7)-manifolds by a Taubes-type gluing of ASD connections on Eguchi–Hanson local models to a flat orbifold connection, followed by a delicate perturbation in weighted Hölder spaces to obtain genuine solutions. The authors develop a robust analytic framework, including improved torsion-free Spin(7) estimates, model-space linear theory on $\mathbb{R}^4\times X$ and $X\times X$, and a fixed-point perturbation on the global manifold $M_t$; this yields over $20{,}000$ four-parameter instanton families across $SO(3),SO(4),SO(5),SO(7),SO(8)$. The construction extends prior work by Lewis through a local-model–driven gluing strategy and refined linear theory, enabling a large supply of explicit examples. The results have potential implications for gauge theory in higher dimensions and string theory compactifications, providing concrete instantiations of higher-dimensional instantons and contributing to virtual enumerative program in Calabi–Yau four-folds.
Abstract
We construct $Spin(7)$-instantons on one of Joyce's compact $Spin(7)$-manifolds. The underlying compact $Spin(7)$-manifold given by Joyce is the same as in Lewis' construction of $Spin(7)$-instantons. However, our construction method and the resulting instantons are new. The compact $Spin(7)$-manifold is constructed by gluing a $Spin(7)$-orbifold and certain local model spaces around the orbifold singularities. We construct our instantons by gluing non-flat connections on the local model spaces to a flat connection on the $Spin(7)$-orbifold. We deliver more than $20,000$ new four-parameter families of examples of $Spin(7)$-instantons within the structure groups $SO(3), SO(4), SO(5), SO(7)$, and $SO(8)$.