$$\mathrm{Spin}(7)$$-Instantons from Evolution Equations

In this paper we study $$\mathrm{Spin}(7)$$ -instantons on asymptotically conical $$\mathrm{Spin}(7)$$ -orbifolds (and manifolds) obtained by filling in certain squashed 3-Sasakian 7-manifolds. We construct a 1-parameter family of explicit $$\mathrm{Spin}(7)$$ -instantons. Taking the parameter to infinity, the family (a) bubbles off an ASD connection in directions transverse to a certain Cayley submanifold Z, (b) away from Z smoothly converges to a limit $$\mathrm{Spin}(7)$$ -instanton that extends across Z onto a topologically distinct bundle, (c) satisfies an energy conservation law for the instantons and the bubbles concentrated on Z, and (d) determines a Fueter section, in the sense of Donaldson and Segal, Haydys and Walpuski.