Standard model gauge fields localized on non-Abelian vortices in six dimensions

A brane-world SU(5) grand unified theory model with global non-Abelian vortices is constructed in six-dimensional spacetime. We find a solution with a vortex associated to SU(3) separated from another vortex associated to SU(2). This 3-2 split configuration achieves a geometric Higgs mechanism for SU(5) -> SU(3) x SU(2) x U(1) symmetry breaking. A simple deformation potential induces a domain wall between non-Abelian vortices, leading to a linear confining potential. The confinement stabilizes the vortex separation moduli, and ensures the vorticities of the SU(3) and SU(2) groups are identical. This dictates the equality of the numbers of fermion zero modes in the fundamental representation of SU(3) (quarks) and SU(2) (leptons), leading to quark/lepton generations. The standard model massless gauge fields arc localized on the non-Abelian vortices thanks to a field-dependent gauge kinetic function. We perform fluctuation analysis with an appropriate gauge fixing and obtain a four-dimensional effective Lagrangian of unbroken and broken gauge fields at quadratic order. We find that SU(3) x SU(2) x U(1) gauge fields are localized on the vortices and exactly massless. Complications in analyzing the spectra of gauge fields with the nontrivial gauge kinetic function are neatly worked out by a vector-analysis-like method.