$G_2$-instantons on the ALC members of the ${\mathbb{B}}_7$ family

Abstract

Using co-homogeneity one symmetries, we construct a two-parameter family of non-abelian $G_2$-instantons on every member of the asymptotically locally conical ${\mathbb{B}}_7$-family of $G_2$-metrics on $S^3\times {\mathbb{R}}^4$, and classify the resulting solutions. These solutions can be described as perturbations of a one-parameter family of abelian instantons, arising from the Killing vector-field generating the asymptotic circle fibre. Generically, these perturbations decay exponentially to the model, but we find a one-parameter family of instantons with polynomial decay. Moreover, we relate the two-parameter family to a lift of an explicit two-parameter family of anti-self-dual instantons on Taub-NUT ${\mathbb{R}}^4$, fibred over $S^3$ in an adiabatic limit.

arXiv comment23 pages, 1 figure. v2: We would like to thank an anonymous reviewer for pointing out a simpler proof of the results in section 3.6, forgoing the need for the theory of irregular singular initial value problems to parameterise asymptotic solutions to the ODE

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