This preprint rigorously constructs the moduli space of triple orthogonal Hopf knot solitons on the classical 3-sphere (S³) and derives the geometric origin of the SU(3) color gauge group from topological perspective. Based on the physical intuition of scale self-duality invariance, we define the flag manifold F = U(3)/T (T is the maximal torus of U(3)) to characterize the triple orthogonal color degrees of freedom of quarks. Using the homotopy long exact sequence of fiber bundles, we prove π₃(F) ≅ ℤ, determining the integer-valued topological degree of smooth maps from S³ to F. We further establish the equivalence between the topological degree and the Hopf invariant, ensuring the quantization of color charge. By constructing the moduli space 𝒜 of smooth maps with degree 1, we prove that 𝒜 is the base space of a principal SU(3) bundle, where the natural SU(3) action on 𝒜 exactly corresponds to the color gauge transformation in Quantum Chromodynamics (QCD). The construction perfectly realizes the three-layer orthogonality constraints (topological constraint, local spherical orthogonality, and symplectic orthogonality on the moduli space) without artificial introduction of symmetry groups or free parameters. This work provides a rigorous mathematical foundation for the geometric description of color interactions and lays a new path for the unification of fundamental interactions in physics.
Xinyu Zheng (Tue,) studied this question.