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

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 $U(1)$-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 $U(1)$-bundles that do not arise from principal ${\mathbb{Z}}_k$-bundles sectors (in particular, projecting out sectors with monopole charges).