An index theorem for Z/2-harmonic spinors branching along a graph
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915649472692224 |
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| author | Haydys, Andriy Mazzeo, Rafe Takahashi, Ryosuke |
| author_facet | Haydys, Andriy Mazzeo, Rafe Takahashi, Ryosuke |
| contents | We prove an index formula for the Dirac operator acting on two-valued spinors on a $3$-manifold $M$ which branch along a smoothly embedded graph $Σ\subset M$, and with respect to a boundary condition along $Σ$ inspired by an instance of this setting related to the deformation theory of $\mathbb Z_2$-harmonic spinors. When $Σ$ is a smooth embedded curve, this index vanishes; this was proved earlier by one of us, but the proof here is different and extends to the more general setting where $Σ$ also has vertices. We focus primarily on the Dirac operator itself, but also show how our results apply to more general twisted Dirac operators and to the closely related $\mathbb Z_2$ harmonic $1$-forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15295 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An index theorem for Z/2-harmonic spinors branching along a graph Haydys, Andriy Mazzeo, Rafe Takahashi, Ryosuke Differential Geometry Analysis of PDEs We prove an index formula for the Dirac operator acting on two-valued spinors on a $3$-manifold $M$ which branch along a smoothly embedded graph $Σ\subset M$, and with respect to a boundary condition along $Σ$ inspired by an instance of this setting related to the deformation theory of $\mathbb Z_2$-harmonic spinors. When $Σ$ is a smooth embedded curve, this index vanishes; this was proved earlier by one of us, but the proof here is different and extends to the more general setting where $Σ$ also has vertices. We focus primarily on the Dirac operator itself, but also show how our results apply to more general twisted Dirac operators and to the closely related $\mathbb Z_2$ harmonic $1$-forms. |
| title | An index theorem for Z/2-harmonic spinors branching along a graph |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2310.15295 |