  Recent breakthrough experiments have demonstrated how it is now possible to
explore the dynamics of quantum Hall states interacting with quantum
electromagnetic cavity fields. While the impact of strongly coupled non-local
cavity modes on integer quantum Hall physics has been recently addressed, its
effects on fractional quantum Hall (FQH) liquids -- and, more generally,
fractionalized states of matter -- remain largely unexplored. In this work, we
develop a theoretical framework for the understanding of FQH states coupled to
quantum light. In particular, combining analytical arguments with tensor
network simulations, we study the dynamics of a $\nu=1/3$ Laughlin state in a
single-mode cavity with finite electric field gradients. We find that the
topological signatures of the FQH state remain robust against the non-local
cavity vacuum fluctuations, as indicated by the endurance of the quantized Hall
resistivity. The entanglement spectra, however, carry direct fingerprints of
light-matter entanglement and topology, revealing peculiar polaritonic replicas
of the $U(1)$ counting. As a further response to cavity fluctuations, we also
find a squeezed FQH geometry, encoded in long-wavelength correlations. We
additionally observe that moving to strong cavity field gradients leads to an
instability towards a sliding Tomonaga-Luttinger liquid phase, featuring a
strong density modulation in the gradient direction. Finally, by exploring the
low-energy excited spectrum inside the FQH phase, we identify a new
quasiparticle, the graviton-polariton, arising from the hybridization between
quadrupolar FQH collective excitations (known as gravitons) and light. We
discuss the experimental implications of our findings and possible extension of
our results to more complex scenarios.
