Paper: Spectral Geometry

Spectral Geometry of an Explicit G₂ Metric on a Compact 7-Manifold

Brieuc de La Fournière (2026) Full text (markdown) | Zenodo DOI: 10.5281/zenodo.19893371

Abstract

First explicit numerical computation of Kaluza-Klein spectrum on compact G₂ manifold. Adiabatic decomposition K₇ ≈ K3 × T² × I reduces 7D PDEs to 1D Sturm-Liouville ODEs. All Betti numbers confirmed spectrally: b₀=1, b₁=0, b₂=21, b₃=77. SD/ASD gap in K3 intersection matrix: 2210×.

Key Results

Scalar Spectrum

Quantity Value
Zero mode λ₀ 3.47×10⁻¹³ (machine zero)
Spectral gap λ₁ 0.1244 ± 0.0001
Weyl law λₙ = 0.125n², α = 1.998 (exact: 2.0)

Betti Number Confirmation

Betti Spectral Gap ratio
b₀ = 1 1 zero mode ,
b₁ = 0 no zero 1-forms ,
b₂ = 21 21 near-zero eigenvalues 14,635×
b₃ = 77 77 near-zero eigenvalues ,

Mass Hierarchy (from SD/ASD gap)

Ratio Spectral Exp. Dev.
m₁/m₂ (τ/μ) 16.5 16.82 1.9%
m₁/m₃ (τ/e) 3400 3477 2.2%
SD/ASD gap 2210× ( )

Adiabatic Validation (5 tests)

Test Result
Fiber flatness < 0.002% max s-variation
Additivity error 0.003–0.023%
Weyl law exponent α = 1.998 (exact: 2.0)
T² isotropy |g^θθ − g^ψψ| = 3×10⁻⁷
K3 roundness spread < 0.1%

KK Tower

Section Structure

  1. Introduction: Context, adiabatic ansatz validation
  2. The Metric: Chebyshev-Cholesky summary, certification
  3. Scalar Laplacian: Spectral gap, Weyl law, KK tower
  4. Hodge Laplacian on 2-Forms, b₂=21 confirmation, SD/ASD structure
  5. Harmonic Forms & Betti Numbers: K3 forms, K₇ assembly, b₃=77
  6. 1-Form Hodge Laplacian: Spectral democracy to 10⁻⁴, b₁=0
  7. Singular Limits: ADE singularity model, spectral stability
  8. Discussion: G₂-MSSM, F-theory, string landscape
  9. Conclusion

Figures

  1. Metric profiles: neck transition and ACyl decay
  2. Scalar eigenvalue staircase (Weyl law)
  3. First 5 scalar eigenfunctions
  4. T² channel spectra (adiabatic additivity)
  5. 2-form spectrum with 14,635× gap