Generalised Wilson Lines

• Generalised Wilson Lines are a unified formalism that extends traditional Wilson lines by incorporating spin, multipole, and geometric effects across various fields.
• They refine gauge coupling along arbitrary and piecewise linear paths using operator-valued holonomies, enabling precise analysis of non-local and topological properties.
• The framework offers practical tools for factorization tests, soft theorem exponentiation, and calculating topological invariants in diverse physical contexts.

Generalised Wilson Lines (GWLs) are a unified formalism for extending the concept of Wilson lines—path-ordered exponentials that encode parallel transport and gauge invariance in quantum field theory—to capture a broader range of local and global structures, including spin, multipole, and geometric effects, in gauge theories, gravity, condensed matter systems, and noncommutative geometry. The GWL formalism refines the coupling along paths to encompass internal degrees of freedom, non-localities, and manifold generalizations, and provides a systematic tool for constructing, analyzing, and evolving operator-valued holonomies under quantum and classical dynamics.

1. Fundamental Definitions and Generalizations

The standard Wilson line along a path $C$ is defined as

$$W[C] = \mathcal{P} \exp \bigg(i g \int_C A_\mu(x)\,dx^\mu \bigg)$$

where $A_\mu(x)$ is a gauge field and $\mathcal{P}$ denotes path ordering, necessary due to non-commutativity in gauge groups.

Generalised Wilson Lines systematically extend this definition by:

• Spin-coupling: Incorporating Pauli terms $F^{\mu\nu}[\gamma_\mu,\gamma_\nu]$ to account for direct spin-field interactions (Cherednikov et al., 2010, Cherednikov et al., 2011). This takes the form

$$\mathcal{P} \exp \left\{ -i g \int_C d z^\mu\, A_\mu^a(z)t^a -i g \kappa \int_C d s\, F^{\mu\nu}_a(z(s)) J_{\mu\nu}\, t^a \right\}$$

where $J_{\mu\nu}=\frac14[\gamma_\mu,\gamma_\nu]$ is the spin generator and $\kappa$ controls coupling strength.

• Multipole and distributional extensions: For composite or extended sources, GWLs average the coupling with the gauge field over spatial distributions, e.g.,

$$\widehat W_{v,\rho}(x) = \exp\left[ i Z e \int_0^\infty ds \int d^3y\, \rho(\mathbf y-\mathbf x)\, v^\mu A_\mu(\mathbf y, v\cdot x+s) \right]$$

and resum all moments of form factors (Plestid, 13 May 2024).

• Gravitational and soft-expansion generalizations: In gravity, replacing $gA_\mu \rightarrow \kappa/2\, h_{\mu\nu}p^\nu$ and adding subleading insertions (derivative and spin) yields GWL operators capable of exponentiating next-to-soft theorems for both scalars and spinning bodies (Bonocore et al., 20 Dec 2024, Bonocore et al., 2021, Fernandes et al., 7 Nov 2025).
• Condensed matter (band geometry): GWLs encode the non-Abelian parallel transport among multi-band Bloch states via the Wilczek–Zee connection, leading to band-geometric holonomies (Li et al., 2015).
• Generalized loop spaces (GLS): GWLs in GLS manipulate functionals on infinite-dimensional spaces of loops, using Fréchet-type shape derivatives to analyze evolution under geometric deformations, especially for light-like polygons in QCD (Cherednikov et al., 2014).
• Noncommutative geometry: In algebraic frameworks, GWLs are defined using covering projections, module paralleltransport, and automorphism groups to generalize holonomy independently of path topology (Ivankov, 2014).

2. Piecewise Linear Decomposition and Diagrammatics

GWLs along arbitrary smooth or piecewise-linear paths decompose into ordered products of straight-segment operators: $W[C] = W[C_M]\, W[C_{M-1}]\cdots W[C_1]$ with each $C_i$ parametrized as $z_i(\lambda)=a_i+\lambda\,\hat n_i$.

For each segment, kernel integrals $I_n$ are also computed: | Segment Type | Kernel $I_n$ Expression | Properties | |------------------|--------------------------------------------------------------------------------|---------------------------------| | Semi-infinite | $$\hat n^{\mu_1}\cdots \hat n^{\mu_n} e^{i a\cdot \sum k_j} \prod_{j=1}^n \frac{i}{\hat n\cdot \sum_{l=j}^n k_l + i \eta}$$ | Building block for all lines | | Finite | Recursive formula; sums over all partitions and positions in segment | Expressed via products of semi-infinite| | Reverse | Hermitian conjugate under path reversal, flips sign and generator order | Color-reversal identities |

Reversing orientation or segment order yields color factors of opposite sign and reversed generator order: $$W_{\text{up}}(t^{a_n}\dots t^{a_1}) F_{a_1\dots a_n} = (-)^n W_{\text{down}}(t^{a_1}\dots t^{a_n}) F_{a_n\dots a_1}.$$ This enables minimal-effort switching between alternate Wilson-line topologies, crucial for testing factorization, universality, and sign-flip relations (e.g., Sivers function sign change).

3. Quantum Corrections, Spin Coupling, and RG Structure

In TMD factorization, GWLs with spin Pauli insertions leave leading-twist distributions unchanged (as $F^{\mu\nu}$ carries one extra derivative and does not contribute at order $1/Q$), but induce nontrivial matrix-valued anomalous dimensions at twist-three. An explicit one-loop UV counterterm is

$$\delta Z_{tw-3} = 1 + \frac{\alpha_s C_F}{4\pi}\frac{1}{\epsilon} [\gamma^+, \gamma^-] + \mathcal{O}(\alpha_s^2)$$

while the RG equation becomes

$$\mu \frac{d}{d\mu} GWL[C] = -\left(\Gamma_{\text{cusp}}[C] + \kappa \Gamma_{\text{spin}}[C]\right) GWL[C]$$

where $\Gamma_{\text{spin}}$ encodes spin-dependent anomalous dimensions (Cherednikov et al., 2010, Cherednikov et al., 2011).

In gravitational theories, GWLs exponentiate all eikonal and next-to-soft emission structures. For spin-½, worldline supersymmetry yields additional $$\sigma^{\nu\sigma} p^\mu \partial_\sigma h_{\mu\nu}(x)$$ terms, corresponding to universally exponentiated next-to-soft amplitudes (Bonocore et al., 20 Dec 2024).

4. Applications: Soft Theorems, Topology, and Multiparticle Correlations

The GWL formalism systematically generates:

• Soft/Next-to-soft theorems: In gravity, soft graviton emission from amplitudes is encoded as exponentiation along worldlines, with connected multi-graviton insertions and matching to universal soft-dressing operators (Fernandes et al., 7 Nov 2025, Bonocore et al., 2021). The inclusion of spin leads to direct classical observables in high-energy scattering.
• Factorization in QCD and beyond: Piecewise-linear GWLs enable minimal diagrammatics for factorization, and universality tests across processes (e.g. SIDIS vs Drell–Yan (Veken, 2014)).
• Gauge-invariant dressings for asymptotic states: In QED and celestial holography, GWLs encode leading and subleading soft conformal dressings, rendering the S-matrix IR finite and directly constructing celestial CFT Ward identities (Nguyen et al., 2023).
• Topological invariants: In band theory, the non-Abelian GWLs encode Berry connection holonomies for multiband systems, yielding direct measurements of Chern, $Z_2$, and topological invariants (Li et al., 2015).
• Noncommutative parallel transport and holonomy: Algebraic GWLs replace continuous loops with automorphism flows, yielding group homomorphisms from covering groups to automorphism groups of modules, and reproducing quantum holonomy in the noncommutative setting (Ivankov, 2014).

5. Evolution Equations, Loop Space Geometry, and Renormalization

Within generalized loop spaces (GLS), variations of cusped lightlike Wilson polygons are governed by Fréchet derivatives, interpreted as infinitesimal diffeomorphisms of loops. The induced evolution equations for Wilson functionals are: $$\left( S_{12}\frac{\partial}{\partial S_{12}} + S_{23}\frac{\partial}{\partial S_{23}} \right) W_* = D_V W_*$$ with scale and rapidity evolution being directly related to shape deformation operators. For $N$ cusps, UV and rapidity log divergences sum as

$$\mu \frac{d}{d\mu} D_V W_* = - \sum_{\text{cusps}} \Gamma_{\text{cusp}}(\alpha_s)$$

$\frac{d}{d\eta} \ln \Phi = -\Gamma_{\text{cusp}}$

where $\Gamma_{\text{cusp}}$ regulates double logarithms at cusp points (Cherednikov et al., 2014).

In multi-line systems, the non-Abelian exponentiation theorem ensures that web diagrams exponentiate into color-connected graphs, with soft anomalous dimension matrices decomposed into minimal bases of connected color structures (Gardi et al., 2013).

6. Extensions to Special and Novel Contexts

GWL concepts extend to:

• NQ-manifolds and higher geometry, with holonomy operators acting on graded bundles equipped with homological vector fields, and representations up to homotopy. Wilson loops become BRST-closed, homotopy, and reparametrization invariant (Bonechi et al., 2011).
• Lattice implementations, augmenting links with clover-type field strength insertions for explicit spin/gauge-covariant dynamics (Cherednikov et al., 2011).
• Worldline Quantum Field Theory (WQFT) and Heavy Effective Theory (HEFT), where GWLs interpolate between off-shell worldline integrals (with classical spin mapping from Grassmann variables) and on-shell amplitude kernels, allowing full resummation of post-Minkowskian expansions in gravity (Bonocore et al., 20 Dec 2024).
• Extended charge distributions and nuclear physics, where GWLs encode radiation and form-factor effects at leading order, subsuming all moments of the charge distribution for coherent processes involving large $Z$ systems (Plestid, 13 May 2024).

7. Open Problems and Future Directions

Current open problems and generalizations include:

• Systematic power counting for segment length in transverse deformations, convergence analysis for fractal/highly-curved paths.
• Extension to operator-valued and non-commutative paths (lattice QCD, non-commutative geometry), and investigation of higher-order corrections (beyond eikonal approximation, subleading soft expansions).
• Exploration of new quantum invariants in generalized geometric, topological, and supermanifold contexts, and applications to quantum computing or synthetic gauge fields.

Generalised Wilson Lines provide a robust algebraic, geometric, and diagrammatic toolkit for encoding nonlocal, internal, and topological properties of gauge, gravitational, and band-theoretic systems. Their continuous development connects perturbative quantum field theory, topological field theory, condensed matter, and noncommutative geometry under a single analytical paradigm.