“Topological” formulation of effective vortex strings Original Research Article

We present a “topological” formulation of arbitrarily shaped vortex strings in four dimensional field theory. By using a large Higgs mass expansion, we then evaluate the effective action of the closed Abrikosov-Nielsen-Olesen vortex string. It is shown that the effective action contains the Nambu-Goto term and an extrinsic curvature squared term with negative sign. We next evaluate the topological FμνF̃μν term in the case where the vortex string manifold extends out to the boundary of the time direction of space-time and find that it becomes the sum of an ordinary self-intersection number and Polyakovʹs self-intersection number of the world sheet swept by the vortex string. These self-intersection numbers are related to the self-linking number and the total twist number, respectively. Furthermore, the FμνF̃μν term turns out to be the difference between the sum of the writhing numbers and the linking numbers of the vortex strings at the initial time and the one at the final time. When the vortex string is coupled to fermions, the chiral fermion number of the vortex string becomes the writhing number (modulo Z) through the chiral anomaly. Our formulation is also applied to “global” vortex strings in a model with a broken global U(1) symmetry.