Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915124525137920 |
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| author | Parker, Gregory J. |
| author_facet | Parker, Gregory J. |
| contents | A $\mathbb Z_2$-harmonic spinor on a 3-manifold $Y$ is a solution of the Dirac equation on a bundle that is twisted around a submanifold $\mathcal Z$ of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of $\mathbb Z_2$-harmonic spinors over the space of parameters $(g,B)$ consisting of a metric and perturbation to the spin connection. The main result states that near a $\mathbb Z_2$-harmonic spinor with $\mathcal Z$ smooth, the universal moduli space projects to a codimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set $\mathcal Z$, necessitating the use of the Nash-Moser Implicit Function Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06245 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds Parker, Gregory J. Differential Geometry Analysis of PDEs Geometric Topology 53, 58, 35 A $\mathbb Z_2$-harmonic spinor on a 3-manifold $Y$ is a solution of the Dirac equation on a bundle that is twisted around a submanifold $\mathcal Z$ of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of $\mathbb Z_2$-harmonic spinors over the space of parameters $(g,B)$ consisting of a metric and perturbation to the spin connection. The main result states that near a $\mathbb Z_2$-harmonic spinor with $\mathcal Z$ smooth, the universal moduli space projects to a codimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set $\mathcal Z$, necessitating the use of the Nash-Moser Implicit Function Theorem. |
| title | Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds |
| topic | Differential Geometry Analysis of PDEs Geometric Topology 53, 58, 35 |
| url | https://arxiv.org/abs/2301.06245 |