Spinor Phases & Structures¶

Detailed treatment of spinor phases and the 64 spinor structures on the Klein quartic.

Spinor Phases¶

On the Klein quartic curve, the spinor phases are determined by the automorphism group \(\mathrm{PSL}(2, 7)\):

Phase	Value	Decimal	Source
\(\delta_A\)	\(\pi/2\)	1.570796	Order-2 element
\(\delta_B\)	\(\pi/3\)	1.047198	Order-3 element
\(\delta_C\)	\(\pi/7\)	0.448799	Order-7 element

Spinor Structures¶

A spinor structure on a Riemannian manifold \((M, g)\) is a lift of the structure group \(\mathrm{SO}(n)\) to \(\mathrm{Spin}(n)\) in the frame bundle.

Count on the Klein Quartic¶

The number of spinor structures on a genus-\(g\) Riemann surface is:

\[ |\mathrm{Spin}(\Sigma_g)| = 2^{2g} = 2^6 = 64 \]

Trivial Spinor Structure¶

The trivial spinor structure \(\sigma_0\) is the one corresponding to the trivial line bundle. It achieves the minimum first eigenvalue of the squared Dirac operator:

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

This is a consequence of the Lichnerowicz formula:

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

which holds for the trivial spinor structure on any spin Riemannian manifold.

Berry Phase Interpretation¶

The b-C correction can be interpreted as a Berry phase — a geometric phase acquired by spinors under parallel transport around a closed loop in the moduli space of spinor structures:

\[ \Delta_{bC} = \lambda_1(D^2_{\sigma_0}) + \frac{\delta_C^2}{2} = 3.338 + 0.100710 = 3.438710 \]

The Berry phase contribution \(\delta_C^2 / 2 \approx 0.1007\) is the first-order correction from the non-trivial holonomy of the spinor bundle.

Computational Access¶

from src.core.spinor_phases import SpinorPhases

phases = SpinorPhases()
print(f"δ_A = {phases.delta_A:.6f}")   # 1.570796
print(f"δ_B = {phases.delta_B:.6f}")   # 1.047198
print(f"δ_C = {phases.delta_C:.6f}")   # 0.448799
print(f"Berry phase = {phases.berry_phase:.6f}")  # 0.100710