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

Tin Sulejmanpasic, Christof Gattringer

TL;DR

This work presents a lattice formulation of abelian gauge theories obtained by gauging the center symmetry of a non-compact U(1) theory, yielding a Villain-type action with a Z-valued 2-form field that enables compact gauge dynamics, monopole control, and natural θ-terms. By deriving a general worldline/worldsheet dual representation, the authors show how to resolve the complex action problem for bosonic matter and reveal electric–magnetic dualities, including a self-dual point in 4d. The paper applies the framework to 2d gauge-Higgs and CP(N−1) models with θ-terms, 3d theories with constrained monopole charges, and 4d θ-terms with the Witten effect, and extends to PSU(N) with discrete θ-terms, offering a pathway for sign-problem-free simulations and new insights into topological and duality structures in lattice 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 $θ$-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 $θ$ 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 $θ$-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 $θ$-terms on a cubic lattice.

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

TL;DR

This work presents a lattice formulation of abelian gauge theories obtained by gauging the center symmetry of a non-compact U(1) theory, yielding a Villain-type action with a Z-valued 2-form field that enables compact gauge dynamics, monopole control, and natural θ-terms. By deriving a general worldline/worldsheet dual representation, the authors show how to resolve the complex action problem for bosonic matter and reveal electric–magnetic dualities, including a self-dual point in 4d. The paper applies the framework to 2d gauge-Higgs and CP(N−1) models with θ-terms, 3d theories with constrained monopole charges, and 4d θ-terms with the Witten effect, and extends to PSU(N) with discrete θ-terms, offering a pathway for sign-problem-free simulations and new insights into topological and duality structures in lattice 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 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 -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 and the gauge Higgs model in with a term, as well as the gauge Higgs model in with constrained monopole charges. These models are of importance in low dimensional anti-ferromagnets. Further we perform a natural construction of the -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 non-abelian gauge theories and the construction of discrete -terms on a cubic lattice.

Paper Structure

This paper contains 21 sections, 126 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Gauging center symmetry and lattice discretization
  3. Compact vs. non-compact abelian gauge theory
  4. The lattice discretization: Gauge fields
  5. r-form U(1) abelian gauge fields in d dimensions
  6. The lattice discretization: Matter fields with electric and magnetic charge
  7. General form of the dual representation with worldlines and worldsheets
  8. 2d U(1) gauge theories with a theta-term
  9. The models and their duals
  10. Worldlines for the mater field weights
  11. Some comments on gauge choices and the theta-term in 2d
  12. 3d U(1) gauge theory with constrained monopole charge
  13. Defining the constraints for the monopole charge
  14. The theta-terms in four dimensions and the Witten effect
  15. The U(1) gauge theory
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Two examples of gauge-equivalent center-symmetry transformations. The dashed line is the co-dimension 1 hyper-surface $\mathcal{S}$ along the links of the dual lattice. The links where the gauge field is shifted by a constant are marked in red. Notice that the orientation of the links is fixed so that the intersection number between $\mathcal{S}$ and the links is positive. The figure on the left and right are related by a gauge transformation which has its support on the site labeled by a black dot.