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Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder

Taro Kimura, Sanchita Sharma

Abstract

We study the Dirac operator on a finite warped cylinder coupled to a background $U(1)$ gauge field. We identify the intrinsic endpoint operators defining the Atiyah-Patodi-Singer (APS) boundary condition and derive a determinant characterization of the modewise APS spectrum. In the constant-gauge, invertible setting, the endpoint reduced $η$ contributions cancel, so the APS index vanishes. For smooth gauge families, the APS projector becomes discontinuous when a boundary mode crosses zero. We therefore introduce a regularized APS-type family of self-adjoint endpoint conditions that remains continuous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslov framework: for nondegenerate regularization, the zero-mode set coincides with the boundary-zero set, and transverse boundary zeros give isolated regular crossings.

Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder

Abstract

We study the Dirac operator on a finite warped cylinder coupled to a background gauge field. We identify the intrinsic endpoint operators defining the Atiyah-Patodi-Singer (APS) boundary condition and derive a determinant characterization of the modewise APS spectrum. In the constant-gauge, invertible setting, the endpoint reduced contributions cancel, so the APS index vanishes. For smooth gauge families, the APS projector becomes discontinuous when a boundary mode crosses zero. We therefore introduce a regularized APS-type family of self-adjoint endpoint conditions that remains continuous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslov framework: for nondegenerate regularization, the zero-mode set coincides with the boundary-zero set, and transverse boundary zeros give isolated regular crossings.
Paper Structure (58 sections, 18 theorems, 273 equations, 5 figures)

This paper contains 58 sections, 18 theorems, 273 equations, 5 figures.

Table of Contents

  1. Introduction and Overview
  2. Main results.
  3. Organization of the paper.
  4. Acknowledgments.
  5. Dirac operator on a finite two-dimensional warped manifold
  6. Spinor bundle and Dirac operator
  7. Function spaces and sections.
  8. Chirality splitting.
  9. Coupling to a background $U(1)$ gauge field
  10. Bulk spectrum
  11. Mode Hilbert spaces.
  12. Reduction to Heun type
  13. APS boundary conditions on the warped cylinder
  14. Normal vector and boundary Clifford multiplication
  15. Self-adjoint boundary Dirac operator
  16. ...and 43 more sections

Key Result

Proposition 3.1

Fix an allowed Fourier mode $k$ and assume $m=k+A\neq 0$. Let act on $\mathfrak H_k=L^2([0,T],f(t)\,dt;\mathbb{C}^2)$ with domain Then $D_{k,\mathop{\mathrm{APS}}\nolimits}$ is a self-adjoint operator on $\mathfrak H_k$ with compact resolvent. In particular, its spectrum is real, discrete, and of finite multiplicity.

Figures (5)

  • Figure 1: Modewise characteristic function $F_k(\lambda)$ for the APS boundary problem at $\alpha=1$, $A=0.3$, $k=1$, and $T=1.5$. The zeros of $F_k(\lambda)$ are the APS eigenvalues in this mode.
  • Figure 2: Example 1 ($A(s)=s-\tfrac{1}{2}$): tracked branches $\lambda(s,k)$ for representative modes $k$. Boundary-zero locations are marked.
  • Figure 3: Example 2 ($A(s)=3s-\tfrac{1}{4}$): tracked branches $\lambda(s,k)$ for selected modes.
  • Figure 4: Example 3 ($A(s)=\sin(4\pi s)$ on $[\varepsilon,1-\varepsilon]$, $\varepsilon=\tfrac{1}{24}$): tracked branches for the selected modes, with endpoint cutoff ensuring invertibility at the ends.
  • Figure 5: Numerical illustration of Proposition \ref{['prop:liouville']} for the parameter values $\alpha=1$, $A=0.3$, $k=1$, and $T=1.5$, matching the setup of \ref{['subsec:bulk_spectrum']}, with representative spectral parameter $\lambda=2$. The plotted quantity is $10^{8}|f(t)\det\psi(t)-2|$, where $\psi(t)$ is the fundamental matrix normalized by $\psi(0)=I$. Since the exact conserved value is $f(0)=2$, the graph measures only numerical deviation from the Liouville/Wronskian identity. Right: zoomed view on a shorter subinterval showing the oscillatory structure more clearly.

Theorems & Definitions (42)

  • Remark 1
  • Proposition 3.1: Modewise APS operator
  • proof
  • Proposition 3.2: Determinant characterization of the modewise APS spectrum
  • proof
  • Proposition 3.3: Absence of zero modes in the invertible case
  • proof
  • Lemma 4.1: Sign/scale flip implies $\xi$-cancellation
  • proof
  • Proposition 4.2: Vanishing of the APS boundary correction for constant gauge
  • ...and 32 more