Non-Abelian Vortex-String Dynamics from Nonlinear Realization

The dynamics of the non-Abelian vortex-string, which describes its low energy oscillations into the target $D=3+1$ spacetime as well as its orientations in the internal space, is derived by the approach of nonlinear realization. The resulting action correlating these two sectors is found to have an invariant synthesis form of the Nambu-Goto-${\bf C}P^{N-1}$ model actions. Higher order corrections to the vortex actions are presented up to the order of quartic derivatives. General $p$-brane dynamics in terms of the internal symmetry breaking is also discussed.