${\rm Spin}(7)$-Instantons from evolution equations

TL;DR

This paper constructs explicit ${ m Spin}(7)$-instantons on asymptotically conical orbifolds, analyzing their bubbling behavior, convergence properties, and associated Fueter sections, advancing understanding of gauge theory in special holonomy manifolds.

Contribution

It introduces a family of explicit ${ m Spin}(7)$-instantons on asymptotically conical manifolds and studies their bubbling, convergence, and Fueter section properties.

Findings

Constructed explicit ${ m Spin}(7)$-instantons on asymptotically conical orbifolds.

Analyzed bubbling phenomena and energy conservation in instanton families.

Connected instanton behavior to Fueter sections in the sense of Donaldson and Segal.

Abstract

In this paper we study ${\rm Spin}(7)$-instantons on asymptotically conical ${\rm Spin}(7)$-orbifolds (and manifolds) obtained by filling in certain squashed $3$-Sasakian $7$-manifolds. We construct a $1$-parameter family of explicit ${\rm Spin}(7)$-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 ${\rm Spin}(7)$-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.