Spectral Geometry and the One-Loop QED ${\hat{I}}^2$-Function on $S^3\times S^1$

Abstract

We compute the one-loop QED ${\hat{I}}^2$-function coefficient directly from heat kernel data of the twisted Spin${}^c$ Dirac operator on $S^3\times S^1$. Using $\hat{I}\P$-function regularization, the logarithmic scale dependence is encoded in the $a_4$ coefficient of the spectral expansion. The $F_{\hat{I}¼\hat{I}½}{F}^{\hat{I}¼\hat{I}½}$ term in $a_4$ yields exactly ${\hat{I}}^2(e)=e^3/(12\ddot{I}{€}^2)$, 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.

13 pages; accepted for publication in Int. J. Geom. Methods Mod. Phys.; DOI: 10.1142/S0219887826501690; arXiv appeal MOD-70631 approved