Inequalities for magnetic-flux free energies and confinement in lattice gauge theories

Rigorous inequalities among magnetic-flux free energies of tori with varying diameters are derived in lattice gauge theories. From the inequalities, it follows that if the magnetic-flux free energy vanishes in the limit of large uniform dilatation of a torus, the free energy must always decrease exponentially with the area of the cross section of the torus. The latter property is known to be sufficient for permanent confinement of static quarks. As a consequence of this property, a lower bound V(R) ? const · R for the static quark-antiquark potential is obtained in three-dimensional U(N) lattice gauge theory for sufficiently large R.