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arXiv · 2310.15295

An index theorem for Z/2-harmonic spinors branching along a graph

Andriy Haydys·Rafe Mazzeo·Ryosuke Takahashi
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Abstract

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.

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Andriy Haydys, Rafe Mazzeo, Ryosuke Takahashi. 2025-12-03. An index theorem for Z/2-harmonic spinors branching along a graph. https://arxiv.org/abs/2310.15295

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