Loop Action for Lattice U(1) Gauge Theory
  It is showed that the very recently introduced Lagrangian $loop$ formulation of the lattice Maxwell theory is equivalent to the Villain form in 2+1 dimensions. A transparent description of the classical $loop$ action is given in pure geometrical terms for the $2+1$ and $3+1$ dimensional cases. 
J. M. Aroca and H. Fort