Gauge theory on Aloff-Wallach spaces

Abstract

For gauge groups $U(1)$ and $SO(3)$ 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 tri-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.

61 pages, 3 figures

References in corpus (7)

• Stabilizing the Complex Structure in Heterotic Calabi-Yau Vacua
• Deformation theory of nearly Kähler manifolds
• $G_2$-instantons over twisted connected sums: an example
• Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons
• $SU(2{)}^2$-invariant $G_2$-instantons
• Deformations of nearly Kähler instantons
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Cited by in corpus (3)

• Deformations of ${\mathrm{G}}_2$-instantons on nearly ${\mathrm{G}}_2$ manifolds
• Examples of deformed Spin(7)-instantons/Donaldson-Thomas connections
• Families of solutions of the heterotic G${}_2$ system