$G_2$-instantons on resolutions of $G_2$-orbifolds
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866918213717065728 |
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| author | Platt, Daniel |
| author_facet | Platt, Daniel |
| contents | We explain a construction of $G_2$-instantons on manifolds obtained by resolving $G_2$-orbifolds. This includes the case of $G_2$-instantons on resolutions of $T^7/Γ$ as a special case. The ingredients needed are a $G_2$-instanton on the orbifold and a Fueter section over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of $T^7/Γ$, improved control over the torsion-free $G_2$-structure allows to remove this assumption. As an application, we construct a large number of $G_2$-instantons on the simplest example of a resolution of $T^7/Γ$ with hundreds of distinct ones among them. We also construct one new example of a $G_2$-instanton on the resolution of $(T^3 \times \text{K3})/\mathbb{Z}^2_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_10903 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $G_2$-instantons on resolutions of $G_2$-orbifolds Platt, Daniel Differential Geometry 53C07, 53C25, 35J60 We explain a construction of $G_2$-instantons on manifolds obtained by resolving $G_2$-orbifolds. This includes the case of $G_2$-instantons on resolutions of $T^7/Γ$ as a special case. The ingredients needed are a $G_2$-instanton on the orbifold and a Fueter section over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of $T^7/Γ$, improved control over the torsion-free $G_2$-structure allows to remove this assumption. As an application, we construct a large number of $G_2$-instantons on the simplest example of a resolution of $T^7/Γ$ with hundreds of distinct ones among them. We also construct one new example of a $G_2$-instanton on the resolution of $(T^3 \times \text{K3})/\mathbb{Z}^2_2$. |
| title | $G_2$-instantons on resolutions of $G_2$-orbifolds |
| topic | Differential Geometry 53C07, 53C25, 35J60 |
| url | https://arxiv.org/abs/2208.10903 |