Systematic Extraction of Exact Yang-Mills Solutions via Algebraic Tensor Ring Decomposition

• URL: http://arxiv.org/abs/2605.06379v1
• Date: Thu, 07 May 2026 14:58:55 GMT
• Title: Systematic Extraction of Exact Yang-Mills Solutions via Algebraic Tensor Ring Decomposition
• Abstract summary: This paper systematically maps the non-linear partial differential equations (PDEs) of Yang-Mills theory into tractable differential-algebraic systems.<n>The framework provides a methodical approach to characterize the classical solution space of strongly coupled gauge theories.
• Score: 5.478582839093083
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The non-linear nature of Yang-Mills theory presents a challenge for extracting exact classical solutions, which are useful for understanding non-perturbative vacuum structures. In this paper, an algebraic tensor ring decomposition framework is introduced to systematically map the non-linear partial differential equations (PDEs) of Yang-Mills theory into tractable differential-algebraic systems. By promoting static pure-gauge backgrounds to dynamical variables, the reference state acts as a geometric template whose Maurer-Cartan forms generate the algebraic cross-terms necessary to stabilize non-linear self-interactions. To analytically resolve the resulting differential ideals, specific differential-algebraic quotient rings are employed as evaluation tools, and the solution space is organized by an algebraic bifurcation analysis. Applying this framework, three distinct classes of exact solutions are extracted: (i) relativistic $SU(2)$ color waves evaluated over an elliptic quotient ring, where the differential ideal bifurcates into a Decoupled Branch and two Coupled Branches, the latter exhibiting mass gap generation; (ii) dynamical dyonic flux tubes obtained from a time-dependent helical template, where the Gauss law ideal bifurcates the system into Coulomb, Dyonic, and symmetric Meissner branches. In the Meissner branch, an Artinian asymptotic truncation yields Bessel-type exponential screening, stabilized by a temporal dominance condition; and (iii) dynamical $SU(3)$ configurations where the Gauss law ideal bifurcates the solution space into four distinct phases. The non-trivial branches enforce a kinetic cancellation mechanism that maps the amplitude dynamics onto a generalized $x^2y^2$ chaotic oscillator. Across these settings, the framework provides a methodical approach to characterize the classical solution space of strongly coupled gauge theories.

Related papers

• Analytical diagonalization of the open-boundary bosonic Kitaev chain: An asymmetric plane-wave ansatz approach [8.29713077555625]
  We present an alternative, purely algebraic solution that relies entirely on the standard bosonic Bogoliubov transformation.<n>For an $N$-site chain, we propose an asymmetric plane-wave ansatz with unequal left- and right-moving momenta to analytically solve the associated $2Ntimes 2N$ non-Hermitian associated matrix.<n>The construction reveals non-uniquenesses that in special cases exactly mirror the freedom in the local squeezing transformations of the original approach.
  arXiv  Detail & Related papers  (2026-08-30T09:37:14Z)
• Nonlocality-induced critical-length hierarchy from non-Hermitian competition [48.9823527611878]
  In short-range systems formed by two coupled chains, competition leads to the critical-length law $N_csimln D$.<n>Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws.<n>Our results offer a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.
  arXiv  Detail & Related papers  (2026-08-03T18:00:18Z)
• Vector Representation of Exact Soliton Dynamics in Multi-component Nonlinear Schrödinger Systems [0.0]
  We develop a vector formulation of the Hirota bilinear formalism for the completely integrable Manakov system.<n>We show how the underlying vector structure provides a unified description of their nonlinear interactions.
  arXiv  Detail & Related papers  (2026-06-25T21:54:02Z)
• An extensive theory of nonlinearly intercoupled pseudomodes for noise model reduction in circuit QED [0.5599792629509229]
  We provide a nonperturbative and systematically reducible framework for open-system cQED dynamics.<n>We develop the general theory in the Heisenberg picture via a Dyson equation for the retained-mode Green's function.
  arXiv  Detail & Related papers  (2026-05-05T16:32:17Z)
• Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation [56.361076943802594]
  CanonFlow achieves state-of-the-art performance on the challenging GEOM-DRUG dataset, and the advantage remains large in few-step generation.
  arXiv  Detail & Related papers  (2026-02-16T18:58:55Z)
• Solving a Nonlinear Eigenvalue Equation in Quantum Information Theory: A Hybrid Approach to Entanglement Quantification [0.0]
  We present a hybrid analytical and numerical framework for evaluating the geometric measure of entanglement.<n>We make the coupled nonlinear eigenstructure explicit by proving the equal multiplier stationarity.<n>The resulting hybrid solver reproduces the exact optimum for standard three qubit benchmarks.
  arXiv  Detail & Related papers  (2025-11-14T13:36:47Z)
• Non-Hermitian $\mathrm{sl}(3, \mathbb{C})$ three-mode couplers [0.0]
  We introduce a general $mathrmsl(N,mathbbC)$ framework for arbitrary $N$-mode couplers in classical and quantum regimes.<n>An exact Wei--Norman propagator captures the full dynamics and makes crossing exceptional points explicit.<n>We study the family spanning $mathcalPT$-symmetric and non-Hermitian cyclic couplers, where two exceptional points of order three lie within a continuum of exceptional points of order two.
  arXiv  Detail & Related papers  (2025-10-28T04:04:11Z)
• Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems [0.638421840998693]
  We revisit a class of models for which the odd solutions were largely missed previously in the literature. We present a systematic and unified treatment for the odd and even sectors of these models. We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters.
  arXiv  Detail & Related papers  (2023-09-21T02:04:44Z)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• General Solution and Canonical Quantization of the Conic Path Constrained Second-Class System [0.0]
  We consider the problem of constrained motion along a conic path under a given external potential function. We perform the canonical quantization in a consistent way in terms of the corresponding Dirac brackets. The complete Dirac brackets algebra in phase space as well as its physical realization in terms of differential operators is explicitly obtained.
  arXiv  Detail & Related papers  (2022-02-15T13:46:56Z)
• Exact solutions of interacting dissipative systems via weak symmetries [77.34726150561087]
  We analytically diagonalize the Liouvillian of a class Markovian dissipative systems with arbitrary strong interactions or nonlinearity. This enables an exact description of the full dynamics and dissipative spectrum. Our method is applicable to a variety of other systems, and could provide a powerful new tool for the study of complex driven-dissipative quantum systems.
  arXiv  Detail & Related papers  (2021-09-27T17:45:42Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.