"Topological" Formulation of Effective Vortex Strings

TL;DR

This paper develops a topological framework for describing vortex strings in four-dimensional field theory, deriving their effective actions and linking topological invariants with physical properties, including effects of fermion coupling.

Contribution

It introduces a topological formulation of vortex strings, evaluates their effective action, and relates topological invariants to physical quantities like fermion number and self-linking.

Findings

Effective action includes Nambu-Goto and extrinsic curvature terms.

The topological  term relates to self-intersection and linking numbers.

Coupling to fermions links chiral fermion number to writhing number.

Abstract

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 $\FtF$ term and find that it becomes the sum of an ordinary self-intersection number and Polyakov's 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 $\FtF$ 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 $\bZ$) through the chiral anomaly. Our formulation is also applied to ``global'' vortex strings in a model with a broken global $U(1)$ symmetry.