  The Peierls instability toward a charge density wave is a canonical example
of phonon-driven strongly correlated physics and is intimately related to
topological quantum matter and exotic superconductivity. We propose a method to
realize an analogous photon-mediated Peierls transition, using a system of
one-dimensional tubes of interacting Bose or Fermi atoms trapped inside a
multimode confocal cavity. Pumping the cavity transversely engineers a
cavity-mediated metal--to--insulator transition in the atomic system. For
strongly interacting bosons in the Tonks-Girardeau limit, this transition can
be understood (through fermionization) as being the Peierls instability. We
extend the calculation to finite values of the interaction strength and derive
analytic expressions for both the cavity field and mass gap. They display
nontrivial power law dependence on the dimensionless matter-light coupling.
