string tensions in the three dimensional Abelian Higgs model. These results are completely new. Some conclusions about the new approach developed here and the induction of repulsive forces between BPS due to their quantum fluctuations offered in Section §.4, where further comments on possible generalizations/extensions of this problem to other physical scenarios are elaborated.

One-loop mass shifts of rotationally symmetric planar BPS vortices
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Quantum fluctuations of BPS vortices in the planar Abelian Higgs model
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The Abelian Higgs model describes the minimal coupling between an $U(1)$-gauge field and a complex scalar field in a phase where the gauge symmetry is spontaneously broken. In terms of non-dimensional variables, $x^\mu \rightarrow \frac{1}{e v}x^\mu$, and fields, $\phi\rightarrow v\phi$, $A_\mu \rightarrow v A_\mu$, where $e$ and $v$ are respectively the gauge coupling and the modulus of the vacuum expectation value of the scalar field, the action functional for the AHM in (2+1)-dimensions reads $$S[\phi,A]=\frac{v}{e}\int d^3x \left[ -\frac{1}{4} F_{\mu\nu}F^{\mu \nu} + \frac{1}{2} (D_\mu \phi)^* D^\mu \phi -\frac{\kappa^2}{8} (\phi^* \phi-1)^2 \right] \, \, \, .
\label{action1}$$ The main ingredients entering this formula are: the complex scalar field $\phi=\phi_1+i\phi_2$, the vector gauge potential $A=(A_0,A_1,A_2)$, the covariant derivative $D_\mu \phi = (\partial_\mu -i A_\mu)\phi$ and the field 