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Algebra-Valued Yang-Mills-Type Theories: Polynomial Obstructions and Holonomy Reductions

Primarily human-written textHuman understanding: all partsmath-ph — Mathematical Physics

Contributed by Yuri Ximenes Lecca ↗, Willian Goulart Gomes Velasco

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

We study algebra-valued Yang-Mills-type (AA-YMT) theories over a smooth nn-manifold MM, where connections take values in a general (not necessarily associative, commutative, or Lie) m\mathfrak{m}-graded algebra (A,∗)(A, *) and descend to an associated algebra bundle EA=P×ρA→ME_A = P \times_\rho A \to M over a principal GG-bundle π:P→M\pi : P \to M. We first derive the variational field equations, infinitesimal gauge transformations, and generalized Bianchi identities, showing that affine Ehresmann compatibility forces the vertical image of g\mathfrak{g} into the Lie nucleus NLie(A)\mathcal{N}_{\mathrm{Lie}}(A) of the commutator [x,y]∗=x∗y−y∗x[x,y]_* = x*y - y*x, and proving that non-linear gauge covariance and the differential Bianchi identity are obstructed precisely by the Jacobiator Jac[⋅,⋅]∗\mathrm{Jac}_{[\cdot,\cdot]_*} (leaving principal GG-gauge transformations universally anomaly-free). We then establish polynomial and geometric obstructions in three settings. First, we prove a vanishing theorem for internal-product Chern-Weil and self-dual AA-YMT functionals in weak (2,s)(2,s)-solvable algebras, invariant under (Q,S)(Q,S)-splitting extensions. Second, in reductive Cartan-Yang-Mills theories valued in A=Rk⊕hA = \mathbb{R}^k \oplus \mathfrak{h}, we compare quartic-block against recursive binary-tree parenthesization, proving a strict dimensional cutoff at n≥8n \ge 8 for quartic blocks and an exact Jacobsthal-number cutoff (n−4∉⋃r≥3{2Jr,Jr+1}n - 4 \notin \bigcup_{r \ge 3}\{2 J_r, J_{r+1}\}, plus coframe-rank vanishing for n≥k+5n \ge k + 5) for binary trees. We also catalog the admissible Berger holonomy reductions for 4≤n≤74 \le n \le 7, establishing an on-shell vanishing (torsion-free) versus off-shell survival (torsionful) dichotomy both in 4D for U(2)U(2) and SU(2)SU(2) connections---via e⋏⋊(e⋏⋊Ωω)=12(e⋏⋊e)⋏⋊Ωωe \curlywedge_\rtimes (e \curlywedge_\rtimes \Omega_\omega) = \frac{1}{2}(e \curlywedge_\rtimes e) \curlywedge_\rtimes \Omega_\omega and the first Bianchi identity---and in 7D for G2G_2-connections. Third, we prove an associator-quotient collapse and Jacobiator vertex-dichotomy theorem for odd-dimensional internal AA-Chern-Simons theories, with applications to AA-BFBF theories, octonionic and reductive Yang-Mills-Higgs Proca mass spectra, and reductive dualities.

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Version history

  1. v1 · 30 pagesSubmitted by Yuri Ximenes LeccaInitial depositCurrentNov 12, 2026