Villain action in lattice gauge theory

TL;DR

The paper proves that the Villain lattice gauge action $V_\beta$ arises as the limit of Wilson and Manton actions on a refined carpet (plaquette-subdivided) graph for any compact connected gauge group $G$, extending the cable-graph idea from spin systems to gauge theories and including non-Abelian cases like $SU(3)$. The result hinges on two pillars: a planar-reduction argument that restores sufficient commutativity to factorize interactions over small plaquettes, and a robust heat-kernel convergence analysis for $G$-valued random walks that upgrades distributional limits to uniform, enabling precise total-variation convergence of the carpet-graph measure to the Villain measure. In the Abelian case $G=U(1)$, the framework yields a monotonicity (Ginibre-type) corollary for Wilson loops with respect to plaquette couplings. The work thus provides a unifying, self-contained approach to connecting Wilson/Manton lattice actions to the Villain formulation, with implications for dualities, correlation inequalities, and ultraviolet stability considerations in lattice gauge theory.

Abstract

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.

Figures (2)

• Figure 1: An example of a carpet graph
• Figure 2: Example of $\mathbb{G}$. Boundary edges are $e_1,e_2,e_3,e_9,e_5$. Denoting by $U_i = U(e_i)$, we have that $U(f_1)$ is the conjugacy class of $U_8^{-1}U_4U_5U_{10}^{-1}U_{10}U_1=U_8^{-1}U_4U_5U_1$, $U(f_2)$ is the conjugacy class of $U_4^{-1}U_9$, $U(f_3)$ is the conjugacy class of $U_3U_8 U_2^{-1}$, and $U(f_\infty)$ is the conjugacy class of $U_1^{-1}U_5^{-1}U_9^{-1}U_3^{-1}U_2$. The edge $e_{10}$ borders only $f_1$ and $U(f_1)$ does not depend on $U_{10}$ as per Remark \ref{['rem:one_face']}.

Theorems & Definitions (23)

• Definition 1.1: Carpet graph
• Remark 1.2
• Definition 1.3: Villain action
• Example 1.4: Abelian Villain action
• Theorem 1.5
• Corollary 1.6
• Remark 1.7
• Example 2.2: Wilson action
• Example 2.3: Manton action
• Lemma 2.4