All Forces from Fiber Torsion

Title: All Forces from Fiber Torsion

Author: Alexander Novickis (alex.novickis@gmail.com)

The four fundamental forces of nature --- electromagnetic, strong, weak, and gravitational --- are conventionally described by four independent gauge theories with three separate gauge groups and a gravitational sector bolted on. We show that all four arise as distinct geometric aspects of a single fiber bundle: the Hopf fibration S¹ → S³ → S² and its flag manifold extension F₂ = SU(3)/[U(1) × U(1)], within the Faddeev-Niemi framework governed by L = (κ₂/2)|∂n|² + (κ₄/4)|F|². Electromagnetism is the holonomy of the S¹ fiber (Berry phase), with coupling α = g² and charge quantisation from π₂(S²) = Z. The strong force is the topology of the flag manifold, with confinement from π₁(SU(3)/Z₃) = Z₃ centre vortices and α_s = α^13/30 ≈ 0.1186. The weak force is the symmetry breaking of the S³ total space, with sin²θ_W = 3/13 from the Berger sphere geometry. Gravity is the backreaction of the soliton condensate on the base manifold, producing emergent Einstein dynamics in the infrared with G = α^{21 − (15/2)α} ħc/m_e² (Paper LXX normal form). No separate gauge groups are postulated. The Standard Model structure SU(3) × SU(2) × U(1) emerges from the topology of the target manifold. This paper assembles results from the programme's technical papers into a unified geometric picture and presents the unification table connecting each force to its fiber-geometric origin.

Key results include:

• All four forces emerge from one fiber bundle (the Hopf fibration) governed by one Lagrangian (Faddeev-Niemi)
• EM = holonomy of the S¹ fiber (Berry phase → Maxwell's equations)
• Strong = topology of F₂ (centre vortices → confinement)
• Weak = symmetry breaking of S³ (Berger squashing → sin²θ_W = 3/13)
• Gravity = condensate curvature (backreaction → emergent Einstein equations)
• No separate gauge groups are postulated; the SM gauge structure arises from target space topology

Keywords: physics, unification, forces, topology, fiber bundle, hopf, gauge theory, gravity, electromagnetism, strong force, weak force

Series: Paper CXIV in the Hopf Soliton Programme