Loop Action for Lattice U(1) Gauge Theory

TL;DR

This paper demonstrates the equivalence of a new loop-based Lagrangian formulation of lattice Maxwell theory to the Villain form in 2+1 dimensions and provides a geometric description of the classical loop action.

Contribution

It introduces a loop formulation of lattice Maxwell theory and shows its equivalence to the Villain form, offering a geometric perspective in multiple dimensions.

Findings

Loop formulation is equivalent to Villain form in 2+1 dimensions.

Provides a geometric description of the classical loop action.

Extends the formulation to 3+1 dimensions.

Abstract

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.