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 with non-commutative space coordinates ($[\hat{ x},\hat{ y}]=i\theta$). This led to consider a different set of the twist matrices $\Omega_{ \mu}$ relative to $T^{ 2}_{ NC}$, since the corrisponding set, adopted in previous works, has resulted to be incompatible with the homogeneity of $T^{ 2}_{ NC}$. 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 $r_{ \theta}=\theta_{ C}/\theta_{ NC}\in\mathbb{ Q}\setminus\{ 0\}$, where $\theta_{ C}$ ($\theta_{ NC}$) is the parameter that characterizes the commutative torus $T^{ 2}_{ C}$ ($T^{ 2}_{ NC}$): here $T^{ 2}_{ C}$ 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.
Categories: hep-th