  We propose a scheme for realizing fractional quantum Hall states of light. In
our scheme, photons of two polarizations are coupled to different atomic
Rydberg states to form two flavors of Rydberg polaritons that behave as an
effective spin. An array of optical cavity modes overlapping with the atomic
cloud enables the realization of an effective spin-1/2 lattice. We show that
the dipolar interaction between such polaritons, inherited from the Rydberg
states, can be exploited to create a flat, topological band for a single
spin-flip excitation. At half filling, this gives rise to a photonic (or
polaritonic) fractional Chern insulator -- a lattice-based, fractional quantum
Hall state of light.
