Currents on superconducting strings at finite chemical potential and temperature

We consider the model giving rise to Witten's superconducting cosmic strings at finite fermion chemical potential and temperature. We demonstrate how various symmetries of the Hamiltonian can be used to exactly compute the fermion electric current in the string background. We show that the current along the string is not sensitive to the profiles of the string fields, and at fixed chemical potential and temperature depends only on the string winding number, the total gauge flux through the vortex and, possibly, the fermion mass at infinity.