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An axion cosmic string is known to be a chiral superconductor when the axioncouples to an electrically charged fermion. After the QCD phase transition, aQCD axion string is attached by $N$ domain walls. We would like to elucidatethe fate of massless fermions on a global string after domain walls attachednot only in the axion model but also in general models having string-wallcomposites. We investigate the Dirac equation under various string-wallcomposite backgrounds both in the axion(-like) models and in the ${\cal N}=2$supersymmetry inspired Abelian-Higgs models. We give an answer to theelementary question of whether massless fermions exist, and if so, where theyare localized. The answer depends on fermion/boson masses in the models, andthe massless fermion can be localized either on the string, on one of thedomain walls, or in one of the vacua. We find analytic solutions for thefermion zero mode function by which we prove the existence of the masslessfermion on the string-wall composites. We also show supercurrents flowing alongthe string-wall composites and anomalous electric currents flowing in fromoutside.

Massless fermions and superconductivity of string-wall composites