string like objects are macroscopic and do not require an specific treatment in a quantum framework. Nielsen and Olesen, however, rediscovered identical extended objects in the relativistic Abelian Higgs model, see [@Nielsen1973:npb], and proposed for them to play a r$\hat{\rm o}$le in hadronic physics as dual strings. It is thus clear after the Nielsen-Olesen proposal that in this new framework the vortex filaments are of quantum nature and there is the need of clarifying to what kind of quantum state they correspond. It was later shown by Bogomolnyi [@Bogomolny1976:sjnp; @Prasad1975:prl] that Abrikosov-Nielsen-Olesen vortices, seen in a two dimensional space, belong to a special class of topological solitons when the masses of the scalar and vector particles in the AHM are equal, or, the correlation lengths of scalar and magnetic fields correspond to the critical point between Type I and Type II phases in Ginzburg-Landau superconductors.

It is thus natural to try the understanding of quantum Bogomolnyi-Prasad-Sommerfield planar vortices in the framework of the general quantum theory of solitons. The first successful attempts in this direction were achieved by Vassilevich in [@Vassilevich2003:prd], and Rebhan et al. in [@Rebhan2004:npb], by attacking this problem in the $N=2$ supersymmetric AHM. Almost simultaneously Bordag and 