This work is motivated by the idea that a compact holonomy degree of freedom (a periodic Wilson-line phase) can be more than a technical parameter in gauge theories with a compact direction: it is an axion-like variable by construction, and in broader settings—especially when coupled to dynamical geometry—it can plausibly play a role in cosmological dynamics (including regimes with dark-matter–like behavior) and in programs where effective couplings are viewed as emergent quantities tied to vacuum geometry. The core purpose of the paper is to turn that broad motivation into something quantitatively usable by building a reproducible computational framework for extracting the most delicate, convention-sensitive semiclassical data from holonomy EFTs. In holonomy EFTs with periodic potentials and multiple vacua, one naturally encounters kinks/domain walls interpolating between vacua, and—once a small bias is introduced—configurations relevant to metastability and vacuum decay. Standard treatments typically make it straightforward to compute “bulk” quantities such as the wall tension and a characteristic wall thickness scale, but the physically important “fine” data are much harder to obtain reliably: in particular, the tail normalization of the wall far from its center (and related prefactors) controls long-range wall–antiwall forces and feeds into semiclassical prefactors in tunneling problems. These quantities are notoriously sensitive to normalization conventions and are expensive to recompute from scratch for each new potential. The paper’s main technical move is to recast the domain-wall sector of the holonomy EFT into a quantum-curve formulation. Instead of treating the wall profile solely as a direct ODE problem in real space, the wall problem is packaged into a spectral-curve object whose global analytic structure can be studied and then quantized. The baseline quantization is a Schrödinger-type differential operator. In addition, the work introduces a q-difference (finite-shift) quantization not as a replacement, but as a complementary, genuinely global diagnostic: because the holonomy variable is periodic, comparing differential and difference quantizations provides a nonlocal test of the periodicity/monodromy data in the semiclassical regime. On top of this quantized curve, the paper constructs a complete exact-WKB pipeline. It generates the all-orders WKB hierarchy, organizes the relevant connection/Voros data that capture global continuation properties, and then performs a practical numerical resummation using Borel–Padé–Laplace methods. Importantly, this is presented as an operational computational workflow with explicit stability checks: sensitivity to truncation order, parameter windows, and lateral resummation choices is monitored so that extracted constants are not artifacts of a particular numerical setting. A key technical ingredient is the introduction of a controlled auxiliary deformation that keeps the turning-point structure well behaved; the physically relevant domain-wall limit is then accessed smoothly by taking the appropriate limit of this deformation parameter. The paper keeps this auxiliary deformation conceptually separate from any physical “tilt/bias” used when discussing metastability and thin-wall vacuum decay. A central claim—tested rather than assumed—is that exact-WKB connection data can be used to extract not only the leading “action-like” information corresponding to wall tension, but also normalization constants that determine the wall tail amplitude and thereby long-range interaction prefactors. To make this claim credible, the work adopts a strict “calibrate → transfer” strategy that freezes conventions. First, the entire setup (quantization choices, branch conventions, normalization, and resummation protocol) is anchored on a solvable sine-Gordon/Mathieu benchmark, where the relevant connection constant is known and can be checked independently. This step is used to lock down the normalization of the method itself, ensuring that the pipeline reproduces the expected universal constant in a setting where the answer is not in dispute. Second, with conventions frozen, the identical workflow is transferred—without retuning—to a more structured, two-harmonic Hill model (a genus-1 benchmark) where independent closed-form control results exist for the wall-tail normalization. This is a crucial stress test: in the two-harmonic case the tail normalization is no longer a universal number but becomes parameter dependent, so success requires the method to track genuinely model-dependent prefactor data rather than merely reproduce a famous constant. The reported agreement across a broad parameter scan is used to argue that the extracted constants are controlled by the global analytic (Stokes/Voros) data intrinsic to the quantum curve, not by ad hoc fitting. Beyond the exact-WKB construction, the paper also places the genus-1 case in the language of topological recursion, showing how recursive spectral geometry can generate the perturbative tower associated with the curve. This is not presented as an aesthetic add-on: it points to a scalable way of organizing higher-order data when the curve becomes more complicated (for example, in multi-harmonic potentials where the effective spectral geometry typically increases in complexity). In that sense, the work lays groundwork for a broader program in which holonomy EFT observables are computed by combining quantum-curve quantization, exact-WKB/resummation, and recursive spectral geometry. What the reader gets at the end is a practical, calibrated dictionary and workflow: starting from a holonomy EFT input, one can systematically obtain wall tension and tail scales, and—crucially—extract tail normalizations and related prefactor information that govern long-range wall interactions. The paper also sketches how these outputs plug into the standard thin-wall mapping for metastable vacuum decay, while clearly separating what is already delivered (the calibrated extraction of the relevant semiclassical data in the wall sector) from what would require additional dedicated work (a fully assembled four-dimensional bounce prefactor). From the standpoint of the motivating big picture, the work does not claim to provide a finished dark-matter model or a complete derivation of emergent couplings. Instead, it supplies an essential missing ingredient for such directions: a controlled method for computing the semiclassical and defect-related data of a compact, axion-like holonomy mode in nontrivial periodic potentials—especially those quantities (tail normalizations and prefactors) that tend to dominate uncertainties in realistic applications. The natural next steps suggested by the framework include extending the same calibrated pipeline to multi-harmonic holonomy potentials typical of realistic gauge-theory settings, exploring nontrivial kinetic metrics, and developing higher-rank generalizations (where an effective one-dimensional reduction may emerge along minimal-tension trajectories). In this way, the paper aims to turn an appealing motivation—holonomy as a physically active compact mode, potentially relevant for cosmology and emergent-geometry ideas—into a concrete computational platform that can be iterated, tested, and expanded.
Dariusz Staniszewski (Mon,) studied this question.