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

Abstract

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.