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Spectral Geometry and the One-Loop QED $β$-Function on $S^3 \times S^1$

Lyudmil Antonov

Abstract

We compute the one-loop QED $β$-function coefficient directly from heat kernel data of the twisted Spin$^c$ Dirac operator on $S^3 \times S^1$. Using $ζ$-function regularization, the logarithmic scale dependence is encoded in the $a_4$ coefficient of the spectral expansion. The $F_{μν} F^{μν}$ term in $a_4$ yields exactly $β(e) = e^3/(12π^2)$, independent of $r$, $L$, or background, verifying spectral RG flow without flat-space propagators. The result is independent of the radii of $S^3$ and $S^1$ and of the choice of gauge background, providing a parameter-free consistency check that spectral data on compact manifolds encode renormalization group information. Beyond a mere verification of the coupling flow, this result serves as a non-trivial consistency check of the Spectral Action Principle in a curved background. It demonstrates that universal quantum corrections can be extracted purely from geometric spectral invariants, distinguishing this geometric spectral derivation from momentum-space propagator methods.

Spectral Geometry and the One-Loop QED $β$-Function on $S^3 \times S^1$

Abstract

We compute the one-loop QED -function coefficient directly from heat kernel data of the twisted Spin Dirac operator on . Using -function regularization, the logarithmic scale dependence is encoded in the coefficient of the spectral expansion. The term in yields exactly , independent of , , or background, verifying spectral RG flow without flat-space propagators. The result is independent of the radii of and and of the choice of gauge background, providing a parameter-free consistency check that spectral data on compact manifolds encode renormalization group information. Beyond a mere verification of the coupling flow, this result serves as a non-trivial consistency check of the Spectral Action Principle in a curved background. It demonstrates that universal quantum corrections can be extracted purely from geometric spectral invariants, distinguishing this geometric spectral derivation from momentum-space propagator methods.
Paper Structure (18 sections, 5 theorems, 38 equations)

This paper contains 18 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Physical motivation and interpretation
  3. Connection to the spectral action principle
  4. Conventions and sign choices
  5. Geometric Setup
  6. Metrics and curvature
  7. Gauge connection and Hopf bundle
  8. Twisted Dirac operator
  9. Heat Kernel Expansion and the a4 Coefficient
  10. General structure
  11. Bundle curvature decomposition
  12. Isolating the F2 contribution
  13. Mapping to the beta-Function via zeta-Regularization
  14. Effective action from the spectral zeta function
  15. One-loop correction to the gauge coupling
  16. ...and 3 more sections

Key Result

Lemma 3.1

The coefficient $a_4(P)$ for a twisted Dirac operator on a four-manifold contains the gauge contribution where $\Omega_{\mu\nu}$ is the total connection curvature on the twisted bundle.

Theorems & Definitions (10)

  • Lemma 3.1: Gilkey
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Remark 3.6