  The interactions between light and matter are strongly enhanced when atoms
are placed in high-finesse quantum cavities, offering tantalizing opportunities
for generating exotic new quantum phases. In this work we show that both
spin-orbit interactions and strong synthetic magnetic fields result when a
neutral atom is confined within a ring cavity, whenever the internal atomic
states are coupled to two off-resonant counter-propagating modes. We
diagonalize the resulting cavity polariton Hamiltonian and find characteristic
spin-orbit dispersion relations for a wide range of parameters. An adjustable
uniform gauge potential is also generated, which can be converted into a
synthetic magnetic field for neutral atoms by applying an external magnetic
field gradient. Very large synthetic magnetic fields are possible as the
strength is proportional to the (average) number of photons in each of the
cavity modes. The results suggest that strong-coupling cavity quantum
electrodynamics can be a useful environment for the formation of topological
states in atomic systems.
