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(De)constructing Continuous Gauge Symmetries

Leron Borsten, Hyungrok Kim

TL;DR

The paper addresses the challenge of discretising continuous U(1) gauge theory by constructing a Z_k-based framework T_k that preserves the monopoleless sector of Maxwell theory in the k→∞ limit. It introduces a decomposition A = A^flat + A^♯, a circle-valued field a, and a Z_k gauge structure with an admissible covariant derivative D^{(q)} that couples matter without relying on a canonical flat connection. Through perturbative equivalence, charge and Wilson loop analyses, and higher-form symmetry considerations, the authors demonstrate that T_k reproduces the monopoleless Maxwell theory and can be interpreted as ordinary Maxwell theory with a nonlocal projection operator projecting onto Z_k-bundle sectors. They also connect monopolelessness to a Higgs mechanism that reduces U(1) to Z_k, providing a physical underpinning for the discretisation. The work opens avenues for extending to non-Abelian groups and higher p-form theories, highlighting both mathematical and physical challenges in discretising gauge symmetries beyond the Abelian case.

Abstract

A $(d+1)$-dimensional field theory with a periodic spatial dimension may be approximated by a $d$-dimensional theory with a truncated Kaluza-Klein tower of $k$ fields; as $k\to\infty$, one recovers the original $(d+1)$-dimensional theory. One may similarly expect that $\operatorname U(1)$-valued Maxwell theory may be approximated by $\mathbb Z_k$-valued gauge theory and that, as $k\to\infty$, one recovers the original Maxwell theory. However, this fails: the $k\to\infty$ limit of $\mathbb Z_k$-valued gauge theory is flat Maxwell theory with no local degrees of freedom. We instead construct field theories $\mathcal T_k$ such that, with appropriate matter couplings, the $k\to\infty$ limit does recover Maxwell theory in the absence of magnetic monopoles (but with possible Wilson loops), and show that $\mathcal T_k$ can be understood as Maxwell theory with the insertion of a certain nonlocal operator that projects out principal $\operatorname U(1)$-bundles that do not arise from principal $\mathbb Z_k$-bundles sectors (in particular, projecting out sectors with monopole charges).

(De)constructing Continuous Gauge Symmetries

TL;DR

The paper addresses the challenge of discretising continuous U(1) gauge theory by constructing a Z_k-based framework T_k that preserves the monopoleless sector of Maxwell theory in the k→∞ limit. It introduces a decomposition A = A^flat + A^♯, a circle-valued field a, and a Z_k gauge structure with an admissible covariant derivative D^{(q)} that couples matter without relying on a canonical flat connection. Through perturbative equivalence, charge and Wilson loop analyses, and higher-form symmetry considerations, the authors demonstrate that T_k reproduces the monopoleless Maxwell theory and can be interpreted as ordinary Maxwell theory with a nonlocal projection operator projecting onto Z_k-bundle sectors. They also connect monopolelessness to a Higgs mechanism that reduces U(1) to Z_k, providing a physical underpinning for the discretisation. The work opens avenues for extending to non-Abelian groups and higher p-form theories, highlighting both mathematical and physical challenges in discretising gauge symmetries beyond the Abelian case.

Abstract

A -dimensional field theory with a periodic spatial dimension may be approximated by a -dimensional theory with a truncated Kaluza-Klein tower of fields; as , one recovers the original -dimensional theory. One may similarly expect that -valued Maxwell theory may be approximated by -valued gauge theory and that, as , one recovers the original Maxwell theory. However, this fails: the limit of -valued gauge theory is flat Maxwell theory with no local degrees of freedom. We instead construct field theories such that, with appropriate matter couplings, the limit does recover Maxwell theory in the absence of magnetic monopoles (but with possible Wilson loops), and show that can be understood as Maxwell theory with the insertion of a certain nonlocal operator that projects out principal -bundles that do not arise from principal -bundles sectors (in particular, projecting out sectors with monopole charges).
Paper Structure (14 sections, 33 equations, 1 figure)

This paper contains 14 sections, 33 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Future directions.
  3. Organisation of this paper.
  4. Discretisation of Maxwell theory in the Čech formulation
  5. The failure of a naïve discretisation of Maxwell theory
  6. Monopoleless subsector of Maxwell theory
  7. Discretised Maxwell theory
  8. Consistency checks
  9. Perturbative equivalence
  10. Matter charges
  11. Wilson loops
  12. Higher-form symmetries
  13. Discretised Maxwell theory as ordinary Maxwell theory with nonlocal operator insertion
  14. Monopolelessness from the Higgs mechanism

Figures (1)

  • Figure 1: Correspondence between Maxwell theory and the discretised Maxwell theory $\mathcal{T}_k$