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

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.

47 pages, 1 figure. Version cleaned up. Mistake in section 5 fixed

References in corpus (5)

• Topological Field Theory of Time-Reversal Invariant Insulators
• Generalized Global Symmetries
• "Deconfined" quantum critical points
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• Surface worm algorithm for abelian Gauge-Higgs systems on the lattice

Cited by in corpus (6)

• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Topological aspects of $4$D Abelian lattice gauge theories with the $\theta$ parameter
• DMRG study of the higher-charge Schwinger model and its 't Hooft anomaly
• Fractons on Graphs and Complexity
• Topology and index theorem with a generalized Villain lattice action -- a test in 2d
• Complex Langevin analysis of 2D U(1) gauge theory on a torus with a $\theta$ term