math-ph — Mathematical Physics

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

Contributed by Yuri Ximenes Lecca, Willian Goulart Gomes Velasco

We study algebra-valued Yang-Mills-type ($A$-YMT) theories over a smooth $n$-manifold $M$, where connections take values in a general (not necessarily associative, commutative, or Lie) $\mathfrak{m}$-graded algebra $(A, *)$ and descend to an associated algebra bundle $E_A = P \times_\rho A \to M$ over a principal $G$-bundle $\pi : P \to M$. We first derive the variational field equations, infinitesimal gauge transformations, and generalized Bianchi identities, showing that affine Ehresmann compatibility forces the vertical image of $\mathfrak{g}$ into the Lie nucleus $\mathcal{N}_{\mathrm{Lie}}(A)$ of the commutator $[x,y]_* = x*y - y*x$, and proving that non-linear gauge covariance and the differential Bianchi identity are obstructed precisely by the Jacobiator $\mathrm{Jac}_{[\cdot,\cdot]_*}$ (leaving principal $G$-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 $A$-YMT functionals in weak $(2,s)$-solvable algebras, invariant under $(Q,S)$-splitting extensions. Second, in reductive Cartan-Yang-Mills theories valued in $A = \mathbb{R}^k \oplus \mathfrak{h}$, we compare quartic-block against recursive binary-tree parenthesization, proving a strict dimensional cutoff at $n \ge 8$ for quartic blocks and an exact Jacobsthal-number cutoff ($n - 4 \notin \bigcup_{r \ge 3}\{2 J_r, J_{r+1}\}$, plus coframe-rank vanishing for $n \ge k + 5$) for binary trees. We also catalog the admissible Berger holonomy reductions for $4 \le n \le 7$, establishing an on-shell vanishing (torsion-free) versus off-shell survival (torsionful) dichotomy both in 4D for $U(2)$ and $SU(2)$ connections---via $e \curlywedge_\rtimes (e \curlywedge_\rtimes \Omega_\omega) = \frac{1}{2}(e \curlywedge_\rtimes e) \curlywedge_\rtimes \Omega_\omega$ and the first Bianchi identity---and in 7D for $G_2$-connections. Third, we prove an associator-quotient collapse and Jacobiator vertex-dichotomy theorem for odd-dimensional internal $A$-Chern-Simons theories, with applications to $A$-$BF$ theories, octonionic and reductive Yang-Mills-Higgs Proca mass spectra, and reductive dualities.