Discrete Approximations to U(1) Principal Bundles in Abelian Gauge Theory

TL;DR

This paper constructs discrete gauge theories that approximate continuous U(1) gauge theory, overcoming limitations of naive Z_k discretizations, and recovers Maxwell theory with matter in the limit as k approaches infinity.

Contribution

The authors develop new discrete gauge theories that accurately approximate U(1) gauge theory, including nonlocal operators to project out monopole sectors, improving upon previous Z_k discretizations.

Findings

The constructed theories recover Maxwell theory without monopoles as k→∞.

Naive Z_k discretizations lead to trivial flat Maxwell theory in the limit.

The approach involves nonlocal operators projecting out non-approximable sectors.

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).