# Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles

## TL;DR

This paper presents a novel lattice discretization of abelian gauge theories that incorporates $ heta$-terms and monopoles, offering advantages like electric-magnetic duality, natural matter coupling, and solutions to complex action problems.

## Contribution

It introduces a center symmetry gauging approach leading to a Villain type action, enabling dualities, monopole control, and $ heta$-term implementation in various dimensions.

## Key findings

- Dualization of models with $ heta$-terms demonstrated
- Solution to complex action problem in worldline/worldsheet representation
- Construction of $ heta$-terms in 2D, 3D, and 4D gauge theories

## Abstract

We discuss a particular lattice discretization of abelian gauge theories in arbitrary dimensions. The construction is based on gauging the center symmetry of a non-compact abelian gauge theory, which results in a Villain type action. We show that this construction has several benefits over the conventional $U(1)$ lattice gauge theory construction, such as electric-magnetic duality, natural coupling of the theory to magnetically charged matter in four dimensions, complete control over the monopoles and their charges in three dimensions and a natural $\theta$-term in two dimensions. Moreover we show that for bosonic matter our formulation can be mapped to a worldline/worldsheet representation where the complex action problem is solved. We illustrate our construction by explicit dualizations of the $CP(N\!-\!1)$ and the gauge Higgs model in $2d$ with a $\theta$ term, as well as the gauge Higgs model in $3d$ with constrained monopole charges. These models are of importance in low dimensional anti-ferromagnets. Further we perform a natural construction of the $\theta$-term in four dimensional gauge theories, and demonstrate the Witten effect which endows magnetic matter with a fractional electric charge. We extend this discussion to $PSU(N)=SU(N)/\mathbb Z_N$ non-abelian gauge theories and the construction of discrete $\theta$-terms on a cubic lattice.