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

The model of dual superconductivity has been revisited considering the $U(1)$-gauged Ginzburg-Landau lagrangian density on a non-commutative torus $T_{NC}^2$, according to a new approach, we propose, in dealing with non-commutative space coordinates ($[\widehat{x},\widehat{y}]=i\theta$). This led to consider a different set of the twist matrices ${\mathrm{\Omega }}_{\mu }$ relative to $T_{NC}^2$, since the corrisponding set, adopted in previous works, has resulted to be incompatible with the homogeneity of $T_{NC}^2$. We have also found that the index labelling the twists $n\in \mathbb{Z}$ no longer has the usual physical role. In fact, the energy, suitably rewritten, of the minimum configurations at the point of Bogomolny and the quark color-electric charge $q_e$ depend on $r_{\theta }={\theta }_C\mathrm{/}{\theta }_{NC}\in \mathbb{Q}\setminus \{0\}$, where ${\theta }_C$ (${\theta }_{NC}$) is the parameter that characterizes the commutative torus $T_C^2$ ($T_{NC}^2$): here $T_C^2$ is not characterized only by a $\theta$ null. The quantity $r_{\theta }$ was then found to 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 as a 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 the twisted boundary conditions solutions without solving, as usual, any system of equations such as the BPS. Another novelty is the presence of a double Higgs mechanism regulated by $r_{\theta }$. In this regard, it predicts, in general, a non null mass for the usual massless Goldstone boson and establishes a relation between the mass spectrum and $r_{\theta }$ according to which a variation over time of $r_{\theta }$ involves a quantized variation in time of such masses that will be null when $r_{\theta }$ assumes values outside a certain interval of values.