  We study the zero temperature phase diagram of a class of two-dimensional
SU(N) antiferromagnets. These models are characterized by having the same type
of SU(N) spin placed at each site of the lattice, and share the property that,
in general, more than two spins must be combined to form a singlet. An
important motivation to study these systems is that they may be realized
naturally in Mott insulators of alkaline earth atoms placed on optical
lattices; indeed, such Mott insulators have already been obtained
experimentally, although the temperatures are still high compared to the
magnetic exchange energy. We study these antiferromagnets in a large-N limit,
finding a variety of ground states. Some of the models studied here have a
valence bond solid ground state, as was found in prior studies, yet we find
that many others have a richer variety of ground states. Focusing on the
two-dimensional square lattice, in addition to valence cluster states (which
are analogous to valence bond states), we find both Abelian and non-Abelian
chiral spin liquid ground states, which are magnetic counterparts of the
fractional quantum Hall effect. We also find a "doubled" chiral spin liquid
ground state that preserves time reversal symmetry. These results are based on
a combination of rigorous lower bounds on the large-N ground state energy, and
a systematic numerical ground state search. We conclude by discussing whether
experimentally relevant SU(N) antiferromagnets -- away from the large-N limit
-- may be chiral spin liquids.
