  We consider a toroidal configuration of cosmic string in 3+1 dimensions in an
abelian Higgs model, a compactification of the Nielsen-Olesen string. This
object is classically unstable. We explicitly compute the number of permitted
zero modes for majorana fermions coupled to such a string. As in the case of
indefinitely long strings, there are |n| zero modes for winding number sector
n, and correspondingly, induced fermionic charge n/2 which canbe fractional.
According to a previously proved result, this implies quantum mechanical
stability for objects with odd winding number. The result is of significance to
cosmology in classes of unified theories permitting such cosmic strings.
