Villain action in lattice gauge theory

We prove that Villain interaction applied to lattice gauge theory can be obtained as the limit of both Wilson and Manton interactions on a larger graph which we call the {\em carpet graph.} This is the lattice gauge theory analog of a well-known property for spin $O(N)$ models where Villain type interactions are the limit of $\mathbb{S}^{N-1}$ spin systems defined on a {\em cable graph}. Perhaps surprisingly in the setting of lattice gauge theory, our proof also applies to non-Abelian lattice theory such as $SU(3)$-lattice gauge theory and its limiting Villain interaction. In the particular case of an Abelian lattice gauge theory, this allows us to extend the validity of Ginibre inequality to the case of the Villain interaction.