Localizing Gauge Fields on a Topological Abelian String and the Coulomb Law

The confinement of electromagnetic field is studied in axial symmetrical, warped, 6D World Brane, using a recently proposed topological abelian string vortex solution as background. It was found, that the massless gauge field fluctuations follow 4D Maxwell equations in the Lorenz gauge. The massless zero mode is localized when the thickness of the string-vortex is less than 5% $\beta/4\pi e^{2}$ there are not others localized massless modes. There is also an infinite of non localized massive Fourier modes, that follow 4 dimensional Proca equations with a continuous spectrum. To compute the corrections to the Coulomb potential, a radial cutoff was introduced, in order to achieve a discrete mass spectrum. As main result, a $\frac{1}{R^{2}}$ correction was found for the 4D effective Coulomb law. Although, this correction is milder than the $% \frac{1}{R^{3}}$ expected for flat 5+1 dimensional electromagnetism, the result is in correspondence with the observed behavior of the Coulomb potential at nowadays measurable distances.