N ov 1 99 4 Fermions on the electroweak string ∗

We construct a simple class of exact solutions of the electroweak theory including the naked Z–string and fermion fields. It consists in the Z–string configuration (φ, Z θ), the time and z components of the neutral gauge bosons (Z 0,3 , A 0,3) and a fermion condensate (lepton or quark) zero mode. The Z–string is not altered (no feed back from the rest of fields on the Z–string) while fermion con-densates are zero modes of the Dirac equation in the presence of the Z–string background (no feed back from the time and z components of the neutral gauge bosons on the fermion fields). For the case of the n–vortex Z–string the number of zero modes found for charged leptons and quarks is (according to previous results by Jackiw and Rossi) equal to |n|, while for (massless) neutrinos is |n| − 1. The presence of fermion fields in its core make the obtained configuration a su-perconducting string, but their presence (as well as that of Z 0,3 , A 0,3) does not enhance the stability of the Z–string.