The QED β-function — the running of the electromagnetic coupling constant with energy scale — is usually computed in flat Minkowski space. The calculation uses Feynman diagrams, dimensional regularization, and the assumption of infinite, flat spacetime. The result is the well-known β₁ = e³/(12π²).
The paper derives the same result on S³ × S¹ — a compact curved spacetime where the spatial part is a three-sphere and the time direction is a circle. No Feynman diagrams, no momentum-space integrals. Instead: the heat kernel of the Dirac operator on the curved space, the spectral ζ-function, and functional determinants.
The method computes the one-loop effective action as a function of the background electromagnetic field. The β-function emerges from how the effective action depends on the field strength: the coefficient of the F² term runs with the energy scale set by the curvature of S³ and the circumference of S¹. The result matches the flat-space answer exactly.
The agreement is expected — the β-function is a local quantity, and local physics shouldn't depend on global topology. But the computational methods are completely different. Flat space uses Fourier transforms and momentum integrals. Curved space uses spectral geometry and zeta-function regularization. That two entirely different mathematical frameworks produce the same number is a consistency check on both. The β-function is robust: it doesn't care which mathematical language you use or which spacetime you work on, as long as the local physics is QED. The number is the physics; everything else is the calculation.