Zero modes and anomalies in superconducting strings

Superconductivity in cosmic strings occurs when electrically charged fermions are trapped as massless particles (Jackiw-Rossi zero modes) in the core of a string. Currents are generated when an electric field is applied along the string, or more realistically, when the string moves through a cosmic magnetic field. In realistic models (e.g., those inspired by grand unified theories), the fermion-vortex systems that arise can be quite complicated and the question of whether or not superconductivity occurs is very model dependent. For example, in certain models, mixing between right- and left-moving zero modes gives rise to an effective mass for the fermions on the string. The currents in this case, at least for reasonable values for cosmic magnetic fields, are uninterestingly small. In this paper, we present a simple method for determining the number of true zero modes in a special class of fermion-vortex systems. These results are then applied to a particular particle-physics model based on the gauge group ${\mathrm{E}}_6$. We also consider the possibility that ${\sum }_{\mathrm{left}\mathrm{movers}}$ ${\mathit{q}}^2$≠${\sum }_{\mathrm{right}\mathrm{movers}}$ ${\mathit{q}}^2$ where q is the electromagnetic charge of a zero mode. In this situation, which occurs in ‘‘frustrated’’ as well as global strings, there is a gauge anomaly (and therefore charge nonconservation) in the effective (1+1)-dimensional theory for the fermion-string system. In the presence of an electric field, the string acquires both charge and current. Charge nonconservation on the string is accounted for by an inflow of charge from the world outside the string. However, both charge and current can be screened, either by polarization of the vacuum or by the surrounding plasma.