From Flux Sheets to True Domain Walls: Non-Abelian Vacuum Holonomy EFT on R3,1 ×S1 R with an explicit SU(2) benchmark and an SU(3) worked extension

In typical treatments, holonomy on a circle is used mainly as an order parameter for locating minima of an effective potential (Hosotani mechanism / Wilson-loop dynamics). The present construction shifts the emphasis: holonomy is promoted to a full EFT degree of freedom, with a controlled field-space geometry (derivative terms) and with direct consequences for nonlinear objects and topological responses. As a result, “topological vacuum response” is not an ad hoc add-on, but part of the same computable effective action obtained by integrating out the Kaluza–Klein tower. A key aspect of this viewpoint is that integrating out heavy modes also generates emergent couplings: coefficients in the low-energy action (both in the parity-even and parity-odd sectors) become functions of holonomy, i.e. parameters that “run over field space” and control dynamics and response. The main interpretive shift is that topological response becomes a property of vacuum branches and of the interfaces between branches, rather than merely a “parameter in a given vacuum.” In the non-Abelian case, distinct conjugacy classes of the holonomy can define physically distinct vacuum branches, and a Cartan kink can interpolate between inequivalent minima, realizing a genuine domain wall (as opposed to an Abelian “flux sheet”). This naturally reframes the topological question as: which response is assigned to a given branch, and what is the response jump across a wall? In the same logic, “couplings” cease to be constants: on different branches and at different points in holonomy space, effective coefficients (e.g. those controlling the holonomy field-space metric or the topological sector) can take different values, providing a concrete example of emergent, vacuum- and field-dependent couplings. In this language, anomaly inflow is no longer only a formal consistency condition; it becomes the mechanism that ties a bulk topological response to a measurable defect response. A jump of a holonomy-dependent topological parameter in the 4D effective action fixes the induced Chern–Simons response on the (2+1)D wall worldvolume. At the same time, the analysis sharply separates the physically invariant content (e.g. the fractional part of the Chern–Simons level, robust against local counterterms) from scheme/regulator-dependent integer shifts. This elevates the topological response to a quantized, robust diagnostic of phases and interfaces, rather than a mere convention. From this perspective, the topological sector is naturally interpreted as an emergent topological coupling: a coefficient determined by the physics of the integrated-out modes and exhibiting a controlled jump across interfaces, rather than an arbitrarily chosen angle appended to the Lagrangian. Moreover, the work shows how the response jump admits a microscopic interpretation through spectral flow in the fermionic KK tower and its relation to the parity anomaly localized on the wall. This places “topological vacuum response” in a clear chain: 5D gauge theory → KK reduction → 4D holonomy EFT with parity-odd terms → wall worldvolume theory with Chern–Simons response. The same chain provides the interpretation of emergent couplings: the effective 4D coefficients (kinetic as well as topological) have an unambiguous origin in integrating out the KK tower and can be stress-tested in controlled limits and benchmark truncations. Finally, introducing a controlled “bias” (branch splitting) closes the physical picture: interfaces with a specified topological response can participate not only as static solitons but as elements of metastable vacuum dynamics (thin-wall bubble nucleation) within the same EFT. This links topological response to quantitative questions of phase stability and inter-branch transitions, while also illustrating that the “couplings” governing these processes (wall tension, response jump, derivative coefficients) are best viewed as emergent EFT quantities, dependent on the vacuum branch and on position in holonomy space.