Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913002535518208 |
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| author | Parker, Gregory J. |
| author_facet | Parker, Gregory J. |
| contents | Given a $\mathbb Z_2$-harmonic spinor satisfying some genericity assumptions, this article constructs a 1-parameter family of two-spinor Seiberg-Witten monopoles converging to it after renormalization. The proof is a gluing construction beginning with model solutions on a neighborhood of the $\mathbb Z_2$-harmonic spinor's singular set. The gluing is complicated by the presence of an infinite-dimensional obstruction bundle for the singular limiting linearized operator. This difficulty is overcome by introducing a generalization of Donaldson's alternating method in which a deformation of the $\mathbb Z_2$-harmonic spinor's singular set is chosen at each stage of the alternating iteration to cancel the obstruction components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03682 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds Parker, Gregory J. Differential Geometry Analysis of PDEs Given a $\mathbb Z_2$-harmonic spinor satisfying some genericity assumptions, this article constructs a 1-parameter family of two-spinor Seiberg-Witten monopoles converging to it after renormalization. The proof is a gluing construction beginning with model solutions on a neighborhood of the $\mathbb Z_2$-harmonic spinor's singular set. The gluing is complicated by the presence of an infinite-dimensional obstruction bundle for the singular limiting linearized operator. This difficulty is overcome by introducing a generalization of Donaldson's alternating method in which a deformation of the $\mathbb Z_2$-harmonic spinor's singular set is chosen at each stage of the alternating iteration to cancel the obstruction components. |
| title | Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2402.03682 |