Construction of special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914796480233472 |
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| author | Arai, Masato Baba, Kurando |
| author_facet | Arai, Masato Baba, Kurando |
| contents | We construct special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold by combining the generalized Legendre transform approach and the moment map technique. The generalized Legendre transform approach provides a formulation to construct hyperkähler manifolds and can make their Calabi-Yau structures manifest. In this approach, the Kähler $2$-forms and the holomorphic volume forms can be written in terms of holomorphic coordinates, which are convenient to employ the moment map technique. This technique derives the condition that a submanifold in the Calabi-Yau manifold is special Lagrangian. For the Taub-NUT manifold and the Atiyah-Hitchin manifold, by the moment map technique, special Lagrangian submanifolds are obtained as a one-parameter family of the orbits corresponding to Hamiltonian action with respect to their Kähler 2-forms. The resultant special Lagrangian submanifolds have cohomogeneity-one symmetry. To demonstrate that our method is useful, we recover the conditions for the special Lagrangian submanifold of the Taub-NUT manifold which is invariant under the tri-holomorphic $U(1)$ symmetry. As new applications of our method, we construct special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold which are invariant under the action of a Lie subgroup of $SO(3)$. In these constructions, our conditions for being special Lagrangian are expressed by ordinary differential equations (ODEs) with respect to the one-parameters. We numerically give solution curves for the ODEs which specify the special Lagrangian submanifolds for the above cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09036 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold Arai, Masato Baba, Kurando Mathematical Physics High Energy Physics - Theory We construct special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold by combining the generalized Legendre transform approach and the moment map technique. The generalized Legendre transform approach provides a formulation to construct hyperkähler manifolds and can make their Calabi-Yau structures manifest. In this approach, the Kähler $2$-forms and the holomorphic volume forms can be written in terms of holomorphic coordinates, which are convenient to employ the moment map technique. This technique derives the condition that a submanifold in the Calabi-Yau manifold is special Lagrangian. For the Taub-NUT manifold and the Atiyah-Hitchin manifold, by the moment map technique, special Lagrangian submanifolds are obtained as a one-parameter family of the orbits corresponding to Hamiltonian action with respect to their Kähler 2-forms. The resultant special Lagrangian submanifolds have cohomogeneity-one symmetry. To demonstrate that our method is useful, we recover the conditions for the special Lagrangian submanifold of the Taub-NUT manifold which is invariant under the tri-holomorphic $U(1)$ symmetry. As new applications of our method, we construct special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold which are invariant under the action of a Lie subgroup of $SO(3)$. In these constructions, our conditions for being special Lagrangian are expressed by ordinary differential equations (ODEs) with respect to the one-parameters. We numerically give solution curves for the ODEs which specify the special Lagrangian submanifolds for the above cases. |
| title | Construction of special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2405.09036 |