The work constructs a controlled low-energy effective description for a five-dimensional gauge theory coupled to Kaluza–Klein (KK) gravity on a spacetime of the form R^ (3, 1) × S¹. In this setting the gauge field contains a global degree of freedom—the holonomy (Wilson line) around the compact dimension. While nonlocal from the 5D viewpoint, in four dimensions it behaves as a periodic axion-like scalar that organizes vacuum branches and minima, and in extra-dimensional model building can govern symmetry breaking and the rearrangement of the 4D mass spectrum (e. g. Hosotani / gauge–Higgs–unification–type mechanisms). The aim is to make vacuum selection and vacuum transitions in KK theories quantitatively reliable once gravity is included, i. e. to determine when holonomy-driven branch structure, domain walls, and tunneling rates are genuine predictions rather than fixed-radius artifacts. A central ingredient is to move beyond the rigid-radius approximation. The circle size is promoted to a dynamical field (the radion r (x) ), and the analysis is carried out in the full KK geometry, including 5D gravity and the KK vector (graviphoton). The radius is therefore not an external parameter but a gravitational degree of freedom tied to vacuum energy, the effective 4D Planck scale, and the local KK scale; radius stabilization and holonomy dynamics must be treated jointly, and freezing the radion can mischaracterize minima, barrier heights, and transitions between neighboring branches. This addresses a common pitfall: computing a fixed-radius holonomy potential and only later adding stabilization by hand, which can miscount minima and misestimate barriers, wall tensions, and decay exponents once backreaction is allowed. Technically, the work builds a low-energy EFT in which the holonomy a and radion r are treated on equal footing. After integrating out complete KK towers, one obtains a 4D Einstein-frame two-field sigma model with field-space metric KIJ (a, r) (including kinetic mixing) and a common effective potential Vₑff (a, r). Practical EFT control criteria are emphasized, not only in vacua but also along defect profiles and semiclassical bounce solutions used to assess metastability. A key structural result is a clean split between UV-sensitive local contributions and nonlocal predictive ones. Using Poisson-resummed one-loop determinants, the k = 0 sector captures the 5D-local UV-sensitive part that renormalizes a finite set of counterterms, while winding sectors k ≠ 0 are UV-finite, holonomy-periodic, and fully calculable. As a result, the periodic branch structure and correlated contributions to radion dynamics are fixed by topology and complete KK spectra, with UV data entering only through a finite set of renormalized local parameters. Relative to much of the fixed-radius holonomy EFT literature, the work provides a single controlled two-field framework in dynamical KK gravity that carries the same EFT consistently through vacua, defects, anomaly inflow, and vacuum decay. The novelty is a unified, auditable two-field EFT—with an explicit predictive/nonpredictive split—covering vacua, defects, inflow, and decay within one consistent set of degrees of freedom. Vacua are described as minima in the coupled (a, r) field space rather than minima of a holonomy potential at fixed radius. The analysis systematizes CT-driven versus IR-driven stabilization—CT-driven set primarily by renormalized local counterterms, and IR-driven dominated by an additional infrared module (e. g. a strong sector with an r-dependent confinement scale). It studies defects/domain walls (holonomy kinks and “flux sheets”) as trajectories in (a, r), and shows in linear response that radion relaxation lowers the effective wall tension. The framework also clarifies gauge and gravitational anomaly inflow across defects, including Chern–Simons level quantization and consistency conditions for fractional levels. Finally, it provides an end-to-end semiclassical pipeline for vacuum decay rates, from thin-wall intuition to full O (4) -symmetric bounces and the Coleman–De Luccia regime.
Dariusz Staniszewski (Mon,) studied this question.