A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi–Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi–Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang–Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian–Einstein connections on the Calabi–Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian–Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.