$G_2$-instantons on the ALC members of the $\mathbb{B}_7$ family
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913858695725056 |
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| author | Stein, Jakob Turner, Matt |
| author_facet | Stein, Jakob Turner, Matt |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03886 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $G_2$-instantons on the ALC members of the $\mathbb{B}_7$ family Stein, Jakob Turner, Matt Differential Geometry 53C07, 53C25 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. |
| title | $G_2$-instantons on the ALC members of the $\mathbb{B}_7$ family |
| topic | Differential Geometry 53C07, 53C25 |
| url | https://arxiv.org/abs/2409.03886 |