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

Andriy Haydys, Rafe Mazzeo, Ryosuke Takahashi

TL;DR

The paper develops an analytic framework for Dirac operators on Z2-twisted spinors branching along graphs in 3-manifolds, deriving an index formula for a boundary problem defined by leading coefficients of Z2-harmonic spinors. It extends the known curve-case to admissible graphs by employing iterated-edge calculus, blowups, and cross-sectional analysis on punctured spheres, assembling parametrices and a global Fredholm theory. The index is expressed in terms of indicial-root data at vertices and a global invariant H, enabling applications to the infinitesimal deformation theory of Z2-harmonic spinors. The results provide rigorous tools for understanding deformations of the branching set and associated spinor fields within twisted Dirac theory, with implications for higher-dimensional gauge-theoretic limits and related geometric analysis.

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.

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

TL;DR

The paper develops an analytic framework for Dirac operators on Z2-twisted spinors branching along graphs in 3-manifolds, deriving an index formula for a boundary problem defined by leading coefficients of Z2-harmonic spinors. It extends the known curve-case to admissible graphs by employing iterated-edge calculus, blowups, and cross-sectional analysis on punctured spheres, assembling parametrices and a global Fredholm theory. The index is expressed in terms of indicial-root data at vertices and a global invariant H, enabling applications to the infinitesimal deformation theory of Z2-harmonic spinors. The results provide rigorous tools for understanding deformations of the branching set and associated spinor fields within twisted Dirac theory, with implications for higher-dimensional gauge-theoretic limits and related geometric analysis.

Abstract

We prove an index formula for the Dirac operator acting on two-valued spinors on a -manifold which branch along a smoothly embedded graph , and with respect to a boundary condition along inspired by an instance of this setting related to the deformation theory of -harmonic spinors. When 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 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 harmonic -forms.
Paper Structure (26 sections, 34 theorems, 182 equations, 1 figure)

This paper contains 26 sections, 34 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Graphs
  4. The blowup
  5. The twisted Dirac operator
  6. $\mathbb Z_2$-harmonic spinors
  7. Analysis of $\mathop{\mathrm{\slashed D}}\nolimits$ when $\Sigma$ is an embedded curve
  8. Background for the edge calculus
  9. Indicial roots
  10. The normal operator
  11. The model Bessel operator $B(\mathop{\mathrm{\slashed D}}\nolimits)$
  12. Mapping properties of $\mathop{\mathrm{\slashed D}}\nolimits$
  13. Asymptotics of solutions
  14. The Calderon subspace
  15. Boundary operators
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\Sigma$ be a smooth closed curve, $\mathcal{I}$ the associated twisting real line bundle and $\psi := \{(c_1^i, d_1^i)\}$ a pair of smooth, complex-valued functions on $\Sigma$ on each edge which never vanish simultaneously. These determine a boundary operator ${\mathcal{T}}_\psi$ for $\mathop{

Figures (1)

  • Figure 1: A graph $\Sigma$ and its blowup $M_\Sigma$. The latter is diffeomorphic to the exterior of the bounded domain shown on the right

Theorems & Definitions (63)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Lemma 1
  • proof
  • Proposition 2
  • ...and 53 more