Gauge theory on Aloff–Wallach spaces

Ball, Gavin¹ ; Oliveira, Goncalo²

¹ Department of Mathematics, Duke University, Durham, NC, United States
² Departamento de Matemàtica Aplicada, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Rio de Janeiro-RJ, Brazil

Geometry & topology, Tome 23 (2019) no. 2, pp. 685-743.

Résumé

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.

Détail

DOI : 10.2140/gt.2019.23.685

Classification : 53C07, 53C29, 53C38, 57R57
Keywords: G2 geometry, gauge theory, instantons, Aloff–Wallach spaces, tri-Sasakian, nearly parallel, cocalibrated

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