  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.
