Holonomy and Spectral Asymmetry in Compactified QED: Euler–Heisenberg and Axion Vacuum Responses from a Kaluza–Klein Fermion

The paper presents a simple, fully controlled example showing that one-loop “vacuum optics” in QED can be tuned by the geometry and topology of an extra dimension. It studies five-dimensional QED on a four-dimensional spacetime with one additional dimension compactified to a circle. Two complexified background parameters act as control knobs: the Wilson-line holonomy around the circle (a global gauge background along the compact direction) and the compactification scale set by the circle radius. The radius is a geometric modulus of the compactification and thus belongs to the metric/gravitational sector: changing it amounts to changing the background geometry, and when promoted to a dynamical radion in a four-dimensional effective description it becomes a scalar gravitational degree of freedom controlling that sector. Throughout the paper, however, the radius is treated as a fixed background modulus; promoting it to a dynamical radion field is a natural extension left for future work. Varying the holonomy and the compactification scale shifts the Kaluza–Klein spectrum and thereby tunes the effective vacuum response. In the CP-even sector, the modulus of the one-loop determinant yields a sum of Euler–Heisenberg–type contributions over the entire KK tower. This produces a tunable nonlinear vacuum electrodynamics—including optical effects such as vacuum birefringence—whose coefficients depend on the holonomy and the circle size. In the CP-odd sector, the phase of the one-loop determinant generates a holonomy-dependent “theta”-type response (axion electrodynamics). For strictly constant fields in infinite, flat four-dimensional space it does not modify local propagation, but it is physically meaningful as a global phase on compact Euclidean backgrounds and as a local response at interfaces, or under slow spacetime variation of the holonomy and/or the radius. In practice, the CP-even and CP-odd responses are linked through the same holonomy- and radius-dependent KK spectrum. Both sectors are tied together by a single simple “holonomy response” function, which admits an interpretation as the Poisson kernel on the circle. It simultaneously controls the CP-odd part and the leading KK sums in the CP-even sector, providing a practical bridge for tracking signs and normalizations. The paper also emphasizes the limits of validity of the effective description: tuning the holonomy and the radius can drive the system toward a closing of the mass gap near KK “level crossings,” where one mode becomes parametrically light. In this regime, integrating out the entire KK tower into a finite set of local operators is no longer controlled. Infrared enhancement generates non-analytic momentum dependence, signaling a genuinely nonlocal effective action. The appropriate low-energy description is therefore an EFT in which this light mode is kept explicitly as a dynamical degree of freedom.