Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909369597165568 |
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| author | Fowdar, Udhav |
| author_facet | Fowdar, Udhav |
| contents | We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills condition on $S^1$-invariant Kähler Einstein $6$-manifolds. This allows us to reformulate the (deformed) Hermitian Yang-Mills equations in terms of data on the quotient Kähler $4$-manifold. In particular, we apply this construction to the canonical bundle of $\mathbb{C}\mathbb{P}^2$ endowed with the Calabi ansatz metric to find explicit abelian $SU(3)$ instantons and we show that these are determined by the spectrum of $\mathbb{C}\mathbb{P}^2$. We also find $1$-parameter families of explicit deformed Hermitian Yang-Mills connections. As a by-product of our investigation we find a coordinate expression for its holomorphic volume form which leads us to construct a special Lagrangian foliation of $\mathcal{O}_{\mathbb{C}\mathbb{P}^2}(-3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_02940 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds Fowdar, Udhav Differential Geometry 53C07, 53C38, 53C55 We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills condition on $S^1$-invariant Kähler Einstein $6$-manifolds. This allows us to reformulate the (deformed) Hermitian Yang-Mills equations in terms of data on the quotient Kähler $4$-manifold. In particular, we apply this construction to the canonical bundle of $\mathbb{C}\mathbb{P}^2$ endowed with the Calabi ansatz metric to find explicit abelian $SU(3)$ instantons and we show that these are determined by the spectrum of $\mathbb{C}\mathbb{P}^2$. We also find $1$-parameter families of explicit deformed Hermitian Yang-Mills connections. As a by-product of our investigation we find a coordinate expression for its holomorphic volume form which leads us to construct a special Lagrangian foliation of $\mathcal{O}_{\mathbb{C}\mathbb{P}^2}(-3)$. |
| title | Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds |
| topic | Differential Geometry 53C07, 53C38, 53C55 |
| url | https://arxiv.org/abs/2207.02940 |