  We connect the translation modes of the instanton in the two-dimensional
Abelian Higgs model with local translations of the vortex of the related model
in (3+1) dimensions, the Nielsen-Olesen vortex. In this context these modes
describe collective oscillations of the string. We construct the wave function
of this mode and we derive, via a virial theorem, an effective action for these
oscillations, which is consistent with the action constructed by Nielsen and
Olesen using general arguments. We discuss some aspects of renormalization,
based on a recent computation of one loop corrections to string tension of the
vortex.
