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Aharonov-Casher theorems for Dirac operators on manifolds with boundary and APS boundary condition

Marie Fialová

TL;DR

This work extends the Aharonov–Casher zero-mode theorem for the Dirac operator to 2D manifolds with boundary under the APS boundary condition, covering the plane with holes, a disk with holes, and a sphere with holes via conformal mapping. By decoupling the Dirac equation into holomorphic and antiholomorphic components and enforcing the APS boundary constraints, the authors derive explicit counts of zero modes determined by the total magnetic flux Φ, with spin alignment dictated by the sign of Φ. The plane with holes yields ⌊|Φ|/(2π)⌋ zero modes, the disk with holes yields ⌊Φ/(2π) + 1/2⌋ in magnitude, and the sphere case uses a semi-total flux hatΦ through the holes, giving ⌊hatΦ/(2π) + 1/2⌋ zero modes; all results are consistent with the APS index theorem. The analysis highlights gauge invariance, conformal changes of metric, and the role of boundary conditions in linking spectral data to topological indices.

Abstract

The Aharonov-Casher theorem is a result on the number of the so-called zero modes of a system described by the magnetic Pauli operator in $\mathbb{R}^2$. In this paper we address the same question for the Dirac operator on a flat two-dimensional manifold with boundary and Atiyah-Patodi-Singer boundary condition. More concretely we are interested in the plane and a disc with a finite number of circular holes cut out. We consider a smooth compactly supported magnetic field on the manifold and an arbitrary magnetic field inside the holes.

Aharonov-Casher theorems for Dirac operators on manifolds with boundary and APS boundary condition

TL;DR

This work extends the Aharonov–Casher zero-mode theorem for the Dirac operator to 2D manifolds with boundary under the APS boundary condition, covering the plane with holes, a disk with holes, and a sphere with holes via conformal mapping. By decoupling the Dirac equation into holomorphic and antiholomorphic components and enforcing the APS boundary constraints, the authors derive explicit counts of zero modes determined by the total magnetic flux Φ, with spin alignment dictated by the sign of Φ. The plane with holes yields ⌊|Φ|/(2π)⌋ zero modes, the disk with holes yields ⌊Φ/(2π) + 1/2⌋ in magnitude, and the sphere case uses a semi-total flux hatΦ through the holes, giving ⌊hatΦ/(2π) + 1/2⌋ zero modes; all results are consistent with the APS index theorem. The analysis highlights gauge invariance, conformal changes of metric, and the role of boundary conditions in linking spectral data to topological indices.

Abstract

The Aharonov-Casher theorem is a result on the number of the so-called zero modes of a system described by the magnetic Pauli operator in . In this paper we address the same question for the Dirac operator on a flat two-dimensional manifold with boundary and Atiyah-Patodi-Singer boundary condition. More concretely we are interested in the plane and a disc with a finite number of circular holes cut out. We consider a smooth compactly supported magnetic field on the manifold and an arbitrary magnetic field inside the holes.
Paper Structure (14 sections, 24 theorems, 151 equations)

This paper contains 14 sections, 24 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Dirac operator and the APS boundary condition
  3. Magnetic field and minimal coupling
  4. Problem setup
  5. Main results
  6. Proof for unbounded region with holes
  7. Trace of the eigenfunctions
  8. Proof for the bounded region with holes
  9. Aharonov--Casher on a sphere with holes
  10. The Dirac operator with APS boundary condition in the conformal metric $g^W$
  11. Relation to the index theorem
  12. Conclusion
  13. The Dirac operator under the stereographic projection
  14. Remarks on Möbius transform

Key Result

Lemma 7

Let $D^{\mathrm{APS}}_a$ and $D_{\widetilde{a}}^{\mathrm{APS}}$ be two Dirac operators with the APS boundary condition on $M$ corresponding to magnetic fields with fluxes $\Phi$ and $\widetilde{\Phi}$ respectively, such that where $\Phi_j$ and $\widetilde{\Phi}_j$ are the fluxes through the hole $\Omega_j$, $j\leq N$ of $a$ and $\widetilde{a}$ respectively, and $\Phi_0$ is the flux of a smooth ma

Theorems & Definitions (58)

  • Definition 1
  • Definition 3
  • Remark 4
  • Remark 5
  • Example 6
  • Lemma 7
  • proof
  • Theorem 8
  • Corollary 9
  • proof
  • ...and 48 more