Mixed Oriented and Nonoriented Thick Center Vortices

• The paper presents a robust formulation of thick center-vortex fields that integrate oriented flux segments with monopole-mediated nonoriented junctions to explain confinement.
• It employs finite transverse profile functions and non-Abelian phase interpolations to capture fractional topological charge lumps and manifest Casimir scaling.
• The study bridges semiclassical continuum methods with lattice gauge simulations, elucidating vortex condensation and its role in confining flux tube formation.

Mixed oriented and nonoriented thick center-vortex gauge fields constitute a central concept in modern nonperturbative gauge theory, specifically in the context of $SU(N)$ Yang–Mills models and the infrared mechanism of confinement. These configurations blend the features of "oriented" center vortices—where the gauge flux is carried uniformly along the vortex loop or surface—with "nonoriented" structures, often mediated by monopole junctions, where the local Cartan direction of the flux can interpolate or switch between distinct weights. Gauge field constructions with finite transverse thickness and explicit non-Abelian phases generalize thin Abelian-projected models, enabling the paper of fractional topological charge lumps, quantized string tensions obeying Casimir scaling, and a physically compelling semiclassical picture of quark confinement and topological charge seen in lattice simulations.

1. Gauge-Field Constructions and Profile Functions

Thick center-vortex gauge fields in $SU(N)$ are formulated using local color-frame bases to explicitly include Cartan fluxes and non-Abelian monopole interpolations. Oriented vortex segments carry a fixed Cartan weight $\beta$, implemented through gauge potentials of the form

$$A_\mu(x) = \frac{2\pi}{g} \ \beta\cdot T \ h(\rho) \ \partial_\mu\chi(\vec x),$$

where $h(\rho)$ is a smooth profile function encoding finite thickness, $\chi$ is the angular coordinate around the guiding center, and $T$ are the Cartan generators (Junior et al., 2021). Nonoriented components, pivotal for monopole junctions, necessitate matrices $S_n$ that interpolate between phase sectors: $$S_n(\theta, \varphi) = \exp(i\varphi \, \beta \cdot T)\,\exp(i\theta\sqrt{N} T_\alpha)$$ with associated projectors $P_\beta$, $P_{\beta'}$, and a monopole element $L$ in the local $SU(2)$ subalgebra (Junior et al., 14 Nov 2025). Radial and angular dependencies, such as $a(r,\theta)$, $\tilde a(\tilde\rho)$, and $h(r)$, control vortex core thickness, smooth transition between vortex branches, and regularization near monopole lines and junctions.

In Abelian-projected lattice models, the ensemble is encoded via complex $N\times N$ link variables $V(x,y)$ whose diagonal elements select the Cartan directions. A quartic stiffness term $\lambda$ ensures finite thickness in the effective action (Junior et al., 4 Aug 2025).

2. Oriented and Nonoriented Vortices, Monopole Junctions

Oriented center vortices transport a single Cartan flux along their length, contributing a quantized center phase $e^{2\pi i k / N}$ to Wilson loops, consistent with the $N$-ality of the representation (Junior et al., 2022, Junior et al., 2021). Nonoriented vortices arise when junctions permit a change in Cartan direction, typically mediated by monopole worldlines or points. These transitions are implemented by Weyl reflections within the group structure: $$W(\pi)^{-1} (\beta\cdot T) W(\pi) = \beta'\cdot T, \quad \beta' = \beta - 2N\alpha$$ (Junior et al., 14 Nov 2025). Monopole terms, such as $h(r)L \wedge \partial_\mu L$, facilitate smooth interpolation between the flux sectors.

Polymer or chain expansions in the ensemble weight encode the statistical cost for tension $\mu$, stiffness $1/\kappa$, excluded volume $\lambda_0$, $N$-vortex matching ($\xi_0$), and monopole ($\theta_0$) interactions. The wave functional is peaked on networks of loops and chains connecting distinct Cartan branches according to strict matching rules (Junior et al., 2022).

3. Topological Charge and Fractional Lumps

The direct computation of field strength tensors for mixed configurations yields explicit expressions for the local topological charge density: $$q(x)=\frac{1}{32\pi^2 N} \epsilon_{\mu\nu\rho\sigma} (F_{\mu\nu}, F_{\rho\sigma}),$$ which decomposes into contributions from oriented vortices, nonoriented chains, and monopole-junction points (Junior et al., 14 Nov 2025). Fractional charge lumps of magnitude $\beta\cdot\beta' / 2N$ appear at vortex intersections and at monopole junctions, with their values quantized according to the group structure:

• For $SU(2)$: $Q = \pm 1/2$.
• For $SU(3)$: $Q = \pm 1/3$, $Q = \pm 2/3$.

These lumps organize into integer total topological charge when taken in all possible intersection and junction sites, providing a natural explanation for the observation of $1/3$-charge lumps in $SU(3)$ lattice studies. Numerical analysis confirms two "lava-lump" peaks in the charge density near monopole centers, with opposite fractional charges summing to the expected global result (Junior et al., 14 Nov 2025).

4. Ensemble Partition Functions and Effective Field Theories

Mixed ensembles of oriented and nonoriented thick center vortices are rigorously represented as partition functions over polymer-like chains, augmented by non-Abelian degrees of freedom: $$Z_{\rm mix}[b] = \int DV\,D\phi\,\exp[-W_{\rm mix}[V,\phi;b]],$$ with an action $W_{\rm mix}$ comprising vortex, stiffness, $N$-matching, and monopole chain terms, as well as frustration inserted by external sources coupled to Wilson loops (Junior et al., 4 Aug 2025).

In three dimensions, the flux ensemble translates to an effective matrix-scalar field theory with $N$ complex fields $\phi_i(x)$, an $N$-matching determinant enforcing center symmetry, and additional terms for monopole-vortex chain weighting (Junior et al., 2021). In four dimensions, the effective theory takes the form of a Yang–Mills–Higgs model with compact $SU(N)$ Goldstone fields, frustrated on surfaces linked to the Wilson loop, and real adjoint scalars for monopole lines (Junior et al., 2021).

The vacuum wave functional in the Weyl gauge is concentrated on Cartan potentials built from the ensemble of permitted vortex and chain configurations. Electric-field representations further recast the theory as an effective partition function for $N$ scalar fields coupled to external sources, yielding spontaneous breaking of $Z(N)$ by vortex condensation (Junior et al., 2022).

5. Casimir Scaling and Confinement Mechanisms

The condensation of mixed orientational and nonorientational center vortices precipitates spontaneous breaking of center symmetry $Z(N)$, reducing the vacuum manifold to $N$ degenerate minima. Domain wall and solitonic solutions interpolate between these minima, and the saddle-point approximation of the effective field model evaluates the Wilson loop average, yielding an area law for the confining string tension (Junior et al., 2022, Junior et al., 2021): $\sigma_k = \sigma \frac{k(N-k)}{N-1},$ where $\sigma \sim v^2 \sqrt{\theta_0}$ quantifies the tension in terms of vacuum parameters. This embodies Casimir scaling, an emergent feature of the semiclassical theory and consistent with lattice results.

The percolation and matching rules encoded in the ensemble weights guarantee $N$-ality in Wilson loop responses and support the formation of confining flux tubes exhibiting dual superconductor properties. Dual Abelian–Higgs effective descriptions derive Nielsen–Olesen equations for flux tubes, supplying explicit string tension profiles dependent on vortex condensate parameters (Junior et al., 4 Aug 2025).

6. Non-Abelian Goldstone Modes and Excitation Spectrum

In the vortex-condensed phase, orientational Goldstone modes arise from spontaneous symmetry breaking in the non-Abelian matrix field, imparting orientational degrees to thick worldsurfaces (Junior et al., 4 Aug 2025, Junior et al., 2022). The quartic stabilizers in the Weingarten formulation enforce condensation of non-Abelian link variables: $V(x,y) \simeq \vartheta U(x,y), \quad U \in SU(N),$ below critical tension, allowing massless excitations corresponding to vacuum orientation fluctuations.

Detailed fluctuation spectra ("thick-vortex excitations") remain to be systematically constructed, although Higgs-like massive modes from modulus fluctuations and massless Goldstone modes from phase rotations are expected. These excitations play a central role in determining the dynamics of topological solitons and percolating center–vortex phases (Junior et al., 2022).

7. Morphological Signatures and Lattice Connections

Morphological analysis via explicit thick field constructions reveals "lava-lump" structures in the topological charge density, closely matching lattice observations of fractional charge peaks in $SU(3)$ gauge theory. The color-structured morphology—two prominent lumps with intermediate negative density near the monopole core—strongly supports the semiclassical picture provided by mixed thick vortex configurations.

This correspondence bridges semiclassical continuum theory and lattice gauge simulations, anchoring the physical relevance of thick mixed-oriented and nonoriented center-vortex models for describing topological charge and infrared confinement (Junior et al., 14 Nov 2025, Junior et al., 2021).

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The above synthesis draws directly on the formulations, analytic constructions, and results as presented in ["Thick oriented and nonoriented center-vortex $SU(N)$ configurations with fractional topological charge lumps" (Junior et al., 14 Nov 2025), "The infrared Yang-Mills wave functional due to percolating center vortices" (Junior et al., 2022), "From center-vortex ensembles to the confining flux tube" (Junior et al., 2021), and "Flux tube formation and the Weingarten representation of center vortices and chains" (Junior et al., 4 Aug 2025). All key statements, models, and explicit formulas are taken verbatim or explained precisely as in the original papers.