#### Volume 23, issue 2 (2019)

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Gauge theory on Aloff–Wallach spaces

### Gavin Ball and Goncalo Oliveira

Geometry & Topology 23 (2019) 685–743
##### Abstract

For gauge groups $U\left(1\right)$ and $SO\left(3\right)$ we classify invariant ${G}_{2}$–instantons for homogeneous coclosed ${G}_{2}$–structures on Aloff–Wallach spaces ${X}_{k,l}$. As a consequence, we give examples where ${G}_{2}$–instantons can be used to distinguish between different strictly nearly parallel ${G}_{2}$–structures on the same Aloff–Wallach space. In addition to this, we find that while certain ${G}_{2}$–instantons exist for the strictly nearly parallel ${G}_{2}$–structure on ${X}_{1,1}$, no such ${G}_{2}$–instantons exist for the $3$–Sasakian one. As a further consequence of the classification, we produce examples of some other interesting phenomena, such as irreducible ${G}_{2}$–instantons that, as the structure varies, merge into the same reducible and obstructed one and ${G}_{2}$–instantons on nearly parallel ${G}_{2}$–manifolds that are not locally energy-minimizing.

##### Keywords
G2 geometry, gauge theory, instantons, Aloff–Wallach spaces, tri-Sasakian, nearly parallel, cocalibrated
##### Mathematical Subject Classification 2010
Primary: 53C07, 53C29, 53C38, 57R57
##### Publication
Received: 23 June 2017
Revised: 31 May 2018
Accepted: 18 September 2018
Published: 8 April 2019
Proposed: Simon Donaldson
Seconded: Tobias H Colding, Tomasz Mrowka
##### Authors
 Gavin Ball Department of Mathematics Duke University Durham, NC United States Goncalo Oliveira Departamento de Matemàtica Aplicada, Instituto de Matemática e Estatística Universidade Federal Fluminense Niterói Rio de Janeiro-RJ Brazil