  We study a class of SU(N) Heisenberg models, describing Mott insulators of
fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic
lattice. Based on an earlier semiclassical analysis, magnetic order is
unlikely, and we focus instead on a solvable large-N limit designed to address
the competition among non-magnetic ground states. We find a rich phase diagram
as a function of the filling parameter k, composed of a variety of ground
states spontaneously breaking lattice symmetries, and in some cases also time
reversal symmetry. One particularly striking example is a state spontaneously
breaking lattice rotation symmetry, where the cubic lattice breaks up into
bilayers, each of which forms a two-dimensional chiral spin liquid state.
