although an identical analysis could be easily developed for anti-vortices. We shall denote by $$\psi(\vec{x};N)=\psi_1(\vec{x};N) + i \, \psi_2(\vec{x};N) \hspace{0.5cm},\hspace{0.5cm} V(\vec{x};N)=(V_1(\vec{x};N),V_2(\vec{x};N))$$ the scalar and vector fields of the BPS vortex solutions; the vorticity number, the magnetic charge, will be specified if necessary. Perturbations of these vortex classical solutions in the form $$\widetilde{A}_i(\vec{x};N)= V_i(\vec{x};N) +\epsilon \, a_i(\vec{x}) \hspace{0.5cm},\hspace{0.5cm} \widetilde{\phi}_i(\vec{x};N)=\psi_i(\vec{x};N) + \epsilon\, \varphi_i(\vec{x}) \hspace{0.5cm},\hspace{0.5cm} i=1,2 \label{perturbed}$$ respond to small fluctuations around the topological defects and open a window to observe the behaviour of these objects in the quantum world up to the semi-classical or one-loop order.

The analysis of the physics of the BPS vortex small fluctuations starts by assembling them in a four-component column which we write as the transpose of the four-component field vector $$\xi(\vec{x})=\left( \begin{array}{cccc}a_1(\vec{x}) & a_2(\vec{x}) & \varphi_1(\vec{x}) & \varphi_2(\vec{x}) \end{array} \right)^{\rm t}\, \, .$$ In order to avoid spurious pure gauge fluctuations we impose the background gauge condition $$B(a_k,\varphi,\phi)=\sum_{k=1}^2 \partial_k a_k-(\psi_1 \varphi_2-\psi_2\varphi_1)=0
\label{backgroundgauge}$$ which can be generated as a field equation by adding to the action the following gauge fixing term: $$S^{(GF)}=\frac{1}{2} \int d^3 x [B(a_k,\varphi,\psi)]^2 \, \, .$$ The expansion of the action up to quadratic order in the fluctuations plus the gauge fixing term reads 