Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 068, 13 pages      arXiv:2512.22114      https://doi.org/10.3842/SIGMA.2026.068

Discrete Approximations to $\operatorname{U}(1)$ Principal Bundles in Abelian Gauge Theory

Leron Borsten and Hyungrok Kim
Centre for Mathematics and Theoretical Physics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, Hertfordshire AL10 9AB, UK

Received March 18, 2026, in final form July 06, 2026; Published online July 18, 2026

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).

Key words: Abelian gauge theory; U(1) principal bundles; discrete gauge symmetry; higher-form symmetry.

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