Charge quantisation without compactness: the Higgs and Yukawa mechanisms from matter geometry

Abstract

In the \emph{geometry-first} formulation of gauge theory \citep{Gomes_internal}, Hermitian vector bundles and their covariant derivatives are the primitive objects. Here gauge groups arise only as the automorphism groups of a small set of structured fundamental fibres, and every matter field is a section of a tensor product of the fundamental bundles. This is a constraint on gauge theories. In the abelian sector it enforces charge quantisation, even without compactness: charges count tensor powers of a fundamental line bundle, so they lie on an integral lattice even when the automorphism group is the non-compact $\mathbb{C}^\times$, whereas a symmetry-first theory built on $\mathbb{R}$ admits irrationally related charges. The Standard Model satisfies the constraint, and its two mass-generating mechanisms follow from the fibre geometry. The Higgs mechanism arises from the second fundamental form of the sub-bundle singled out by the Higgs vacuum. This works with no appeal to symmetry breaking or Goldstone's theorem, and the full electroweak spectrum, $m_W=m_Z\cos\theta_W$ included, is computed from the same operator. Each Yukawa coupling is the canonical fibrewise contraction fixed by the inner products and volume forms of the fundamental bundles. Exceptional gauge groups remain formally presentable, but on progressively less elementary fibre structure; at $E_8$ the fundamental fibre is the Lie algebra $\mathfrak{e}_8$ itself, and the claim of conceptual priority collapses. A companion paper shows the two formulations are not equivalent, in a categorical sense \citep{Gomes_nonequiv}.