Topology of the Dirac equation on spectrally large three-manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909062845693952 |
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| author | Lin, Francesco |
| author_facet | Lin, Francesco |
| contents | The interaction between spin geometry and positive scalar curvature has been extensively explored. In this paper, we instead focus on Dirac operators on Riemannian three-manifolds for which the spectral gap $λ_1^*$ of the Hodge Laplacian on coexact $1$-forms is large compared to the curvature. As a concrete application, we show that for any spectrally large metric on the three-torus $T^3$, the locus in the torus of flat $U(1)$-connections where (a small generic pertubation of) the corresponding twisted Dirac operator has kernel is diffeomorphic to a two-sphere. While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin$^c$ three-manifold $(Y,\mathfrak{s})$ with a large spectral gap $λ_1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak{s})$ in terms of the topology of the family of Dirac operators parametrized by the torus of flat $U(1)$-connections on $Y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topology of the Dirac equation on spectrally large three-manifolds Lin, Francesco Differential Geometry Geometric Topology Spectral Theory The interaction between spin geometry and positive scalar curvature has been extensively explored. In this paper, we instead focus on Dirac operators on Riemannian three-manifolds for which the spectral gap $λ_1^*$ of the Hodge Laplacian on coexact $1$-forms is large compared to the curvature. As a concrete application, we show that for any spectrally large metric on the three-torus $T^3$, the locus in the torus of flat $U(1)$-connections where (a small generic pertubation of) the corresponding twisted Dirac operator has kernel is diffeomorphic to a two-sphere. While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin$^c$ three-manifold $(Y,\mathfrak{s})$ with a large spectral gap $λ_1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak{s})$ in terms of the topology of the family of Dirac operators parametrized by the torus of flat $U(1)$-connections on $Y$. |
| title | Topology of the Dirac equation on spectrally large three-manifolds |
| topic | Differential Geometry Geometric Topology Spectral Theory |
| url | https://arxiv.org/abs/2401.02724 |