Dirac Operator¶

The Dirac operator and its spectral properties on the Klein quartic.

Lichnerowicz Formula¶

For any spinor structure \(\sigma\) on a spin Riemannian manifold, the Lichnerowicz formula relates the squared Dirac operator to the Laplacian and scalar curvature:

\[ D^2_\sigma = \Delta + \frac{R}{4} \]

On the Klein quartic with \(R = -2\):

\[ \lambda_1(D^2_{\sigma_0}) = \lambda_1(\Delta) + \frac{R}{4} = 3.838 + (-0.5) = 3.338 \]

First Eigenvalue of the Laplacian¶

The value \(\lambda_1(\Delta) = 3.838\) was rigorously computed by Bourque & Strohmaier (2024) using Selberg trace formula methods on the hyperbolic surface.

Dirac Spectrum¶

The Dirac operator \(D\) on a compact spin Riemannian manifold has a discrete real spectrum symmetric about zero:

\[ \mathrm{Spec}(D_\sigma) = \{ \ldots, -\lambda_2, -\lambda_1, \lambda_1, \lambda_2, \ldots \} \]

For the trivial spinor structure on the Klein quartic, the first positive eigenvalue is:

\[ \lambda_1(D_{\sigma_0}) = \sqrt{3.338} \approx 1.827 \]

Computational Access¶

from src.core.dirac_operator import DiracOperator

dirac = DiracOperator()
print(f"λ₁(Δ) = {dirac.lambda_1_laplacian:.4f}")        # 3.838
print(f"λ₁(D²) = {dirac.lambda_1_dirac_squared:.4f}")   # 3.338
print(f"λ₁(D)  = {dirac.lambda_1_dirac:.4f}")           # 1.827
print(f"R      = {dirac.scalar_curvature}")               # -2