2023/10/23 by Andriy Haydys, Rafe Mazzeo, Haydys, Andriy +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2310.15295
openalex publication_date 2023/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊂ M, and with respect to a boundary condition along Σ inspired by an instance of this setting related to the deformation theory of \mathbb Z2-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 Z2 harmonic 1-forms.