theorem, see also [@Weinberg2012], stated that the algebraic kernel of the deformation operator has dimension $2 N$ where $N$ is the number of quanta of magnetic flux (vorticity) carried by the vortex solution. This means that there exist $2 N$ zero modes of fluctuation around BPS vortices linearly independent. Our main goal in this paper is to compute the quantum correction to the BPS vortex string tension induced by the vortex fluctuations taking into account the existence of these vortex zero modes. Essentially we shall follow an strategy similar to that developed in [@Barvinsky1990:npb] and [@Gusev2000:prd] but we shall adapt our treatment to the heat kernel/zeta function procedure as applied in quantum theory of solitons. Specifically, our new technique is tailored in order to incorporate the impact of zero modes in the infrared in the Gilkey-deWitt heat kernel expansion. In fact, in Reference [@Alonso2012:epjc] we proposed and tested the improved heat kernel expansion, with the impact of zero modes under control, in scalar one-field theory models in order to compute one-loop kink mass shifts. Limitations in the use of the standard GdW procedure arise when zero modes enter the game because the asymptotic low temperature behaviour of the heat function 