  Since the discovery of the Harper-Hofstadter model, it has been known that
condensed matter systems with periodic modulations can be promoted to
non-trivial topological states with emergent gauge fields in higher dimensions.
In this work, we develop a general procedure to compute the gauge fields in
higher dimensions associated to low-dimensional systems with periodic (charge-
and spin-) density wave modulations. We construct two-dimensional (2D) models
with modulations that can be promoted to higher-order topological phases with
$U(1)$ and $SU(2)$ gauge fields in 3D. Corner modes in our 2D models can be
pumped by adiabatic sliding of the phase of the modulation, yielding hinge
modes in the promoted models. We also examine a 3D Weyl semimetal (WSM) gapped
by charge-density wave (CDW) order, possessing quantum anomalous Hall (QAH)
surface states. We show that this 3D system is equivalent to a 4D nodal line
system gapped by a $U(1)$ gauge field with a nonzero second Chern number. We
explain the recently identified interpolation between inversion-symmetry
protected phases of the 3D WSM gapped by CDWs using the corresponding 4D
theory. Our results can extend the search for (higher-order) topological states
in higher dimensions to density wave systems.
