vortex strings. To perform this task, evaluation of the quantum corrections to BPS vortices by using the modified GdW expansion we need a detailed information of the spectrum of the matrix second-order PDO that governs the vortex fluctuations. All the information needed about vortex zero modes and bound states is collected in our recent papers [@Alonso2016:plb] and [@Alonso2016:jhep] where pertinent References can be found.

The organization of the paper is as follows: In Section §.2 we thoroughly address the problem described above in the planar system. Subsection §.2.1 summarize the well known facts about planar BPS fluctuations, in sub-Section §.2.2 the modified or improved GdW heat kernel expansion is generalized to planar Abelian gauge systems with spontaneous symmetry breaking by one scalar field, and finally, in sub-Section §.2.3 the one-loop mass shifts of rotationally symmetric planar BPS vortices are computed. The new one-loop mass vortex shifts performed in this work, although qualitatively compatible with those obtained in [@Alonso2005:prd], are of much greater precision because the new technique is able to incorporate also the effect of zero modes in the spectral zeta function. Section §.3 is fully devoted to describe the quantum corrections at one loop order of the BPS vortex 