Convergence of \(\text{U}(1)_ 3\) lattice gauge theory to its continuum limit (Q1061405)

Lattice approximations of the free Euclidean electromagnetic field in three space-time dimensions are considered, extending results of \textit{K. G. Wilson} [Confinement of quarks, Phys. Rev. D 10, 2445--2450 (1974)]: The compact \(\text{U}(1)_ 3\) lattice gauge theory in three dimensions converges to the electromagnetic field (in the sense of convergence of the characteristic functional of the field variables \(F_{u,v})\) for the Villain action, and for the Wilson action on the electric sector. There are no monopoles when starting with the Villain action, and the same holds starting with the spin wave approximation of the Wilson action, so that it is conjectured that there are no monopoles in the continuum limit of the Wilson action.