$G_2$-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.

23 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

References in corpus (3)

• $SU(2{)}^2$-invariant gauge theory on asymptotically conical Calabi-Yau 3-folds
• $G_2$-instantons on the spinor bundle of the 3-sphere
• Instantons and Bows for the Classical Groups