Spectral Geometry and the One-Loop QED 𝛽-Function on 𝑆³×𝑆¹

1 Introduction

1. 1.1 Physical motivation and interpretation
2. 1.2 Connection to the spectral action principle
3. 1.3 Conventions and sign choices

2 Geometric Setup

1. 2.1 Metrics and curvature
2. 2.2 Gauge connection and Hopf bundle
3. 2.3 Twisted Dirac operator

3 Heat Kernel Expansion and the $a_{4}$ Coefficient

1. 3.1 General structure
2. 3.2 Bundle curvature decomposition
3. 3.3 Isolating the $F^{2}$ contribution

4 Mapping to the $\beta$-Function via $\zeta$-Regularization

1. 4.1 Effective action from the spectral zeta function
2. 4.2 One-loop correction to the gauge coupling
3. 4.3 The $\beta$-function

5 Discussion and Physical Interpretation

1. 5.1 Universality and parameter independence
2. 5.2 Limitations and the UV scale problem
3. 5.3 Comparison with related work