  An axion cosmic string is known to be a chiral superconductor when the axion
couples to an electrically charged fermion. After the QCD phase transition, a
QCD axion string is attached by $N$ domain walls. We would like to elucidate
the fate of massless fermions on a global string after domain walls attached
not only in the axion model but also in general models having string-wall
composites. We investigate the Dirac equation under various string-wall
composite backgrounds both in the axion(-like) models and in the ${\cal N}=2$
supersymmetry inspired Abelian-Higgs models. We give an answer to the
elementary question of whether massless fermions exist, and if so, where they
are localized. The answer depends on fermion/boson masses in the models, and
the massless fermion can be localized either on the string, on one of the
domain walls, or in one of the vacua. We find analytic solutions for the
fermion zero mode function by which we prove the existence of the massless
fermion on the string-wall composites. We also show supercurrents flowing along
the string-wall composites and anomalous electric currents flowing in from
outside.
