  We study the quantum Hall effect in a two-dimensional homogeneous electron
gas coupled to a quantum cavity field. As initially pointed out by Kohn,
Galilean invariance for a homogeneous quantum Hall system implies that the
electronic center of mass (CM) decouples from the electron-electron
interaction, and the energy of the CM mode, also known as Kohn mode, is equal
to the single particle cyclotron transition. In this work, we point out that
strong light-matter hybridization between the Kohn mode and the cavity photons
gives rise to collective hybrid modes between the Landau levels and the
photons. We provide the exact solution for the collective Landau polaritons and
we demonstrate the weakening of topological protection at zero temperature due
to the existence of the lower polariton mode which is softer than the Kohn
mode. This provides an intrinsic mechanism for the recently observed
topological breakdown of the quantum Hall effect in a cavity [Appugliese et
al., Science 375, 1030-1034 (2022)]. Importantly, our theory predicts the
cavity suppression of the thermal activation gap in the quantum Hall transport.
Our work paves the way for future developments in the cavity control of quantum
materials.
