  We investigate long-range pairing interactions between ultracold fermionic
atoms confined in an optical lattice which are mediated by the coupling to a
cavity. In the absence of other perturbations, we find three degenerate pairing
symmetries for a two-dimensional square lattice. By tuning a weak local atomic
interaction via a Feshbach resonance or by tuning a weak magnetic field, the
superfluid system can be driven from a topologically trivial s-wave to
topologically ordered, chiral superfluids containing Majorana edge states. Our
work points out a novel path towards the creation of exotic superfluid states
by exploiting the competition between long-range and short-range interactions.
