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The model of dual superconductivity has been revisited considering the$U(1)$-gauged Ginzburg-Landau lagrangian density on a non-commutative torus$T^{2}_{NC}$, according to a new approach, we propose, in dealing withnon-commutative space coordinates ($[\hat{x},\hat{y}]=i\theta$). This led toconsider a different set of the twist matrices $\Omega_{\mu}$ relative to$T^{2}_{NC}$, since the corrisponding set, adopted in previous works, hasresulted to be incompatible with the homogeneity of $T^{2}_{NC}$. We have alsofound that the index labelling the twists $n\in\mathbb{Z}$ no longer has theusual physical role. In fact, the energy, suitably rewritten, of the minimumconfigurations at the point of Bogomolny and the quark color-electric charge$q_{e}$ depend on$r_{\theta}=\theta_{C}/\theta_{NC}\in\mathbb{Q}\setminus\{0\}$, where$\theta_{C}$ ($\theta_{NC}$) is the parameter that characterizes thecommutative torus $T^{2}_{C}$ ($T^{2}_{NC}$): here $T^{2}_{C}$ is notcharacterized only by a $\theta$ null. The quantity $r_{\theta}$ was then foundto determine the order of a cyclic subgroup of $U(1)$ generated by$e^{i\pi\theta_{NC}}$. Therefore here $q_{e}$ can also be a fraction as well asa multiple of $g^{-1}$, where $g$ is the magnetic charge of the scalar field.Furthermore, the above energy was calculated only considering a class of thetwisted boundary conditions solutions without solving, as usual, any system ofequations such as the BPS. Another novelty is the presence of a double Higgsmechanism regulated by $r_{\theta}$. In this regard, it predicts, in general, anon null mass for the usual massless Goldstone boson and establishes a relationbetween the mass spectrum and $r_{\theta}$ according to which a variation overtime of $r_{\theta}$ involves a quantized variation in time of such masses thatwill be null when $r_{\theta}$ assumes values outside a certain interval ofvalues.

Vortex on a non-commutative torus in the dual superconductivity model: a  link between a cyclic $\tau\subset U(1)$, the color-electric charge of quarks  and the vortex mass