  We propose an experimentally feasible setup with ultracold alkaline earth
atoms to simulate the dynamics of U(1) lattice gauge theories in 2+1 dimensions
with a Chern-Simons term. To this end we consider the ground state properties
of spin-5/2 alkaline earth fermions in a honeycomb lattice. We use the
Gutzwiller projected variational approach in the strongly repulsive regime in
the case of filling 1/6. The ground state of the system is a chiral spin liquid
state with $2\pi/3$ flux per plaquette, which spontaneously violates time
reversal invariance. We demonstrate that due to the breaking of time reversal
symmetry the system exhibits quantum Hall effect and chiral edge states. We
relate the experimentally accessible spin fluctuations to the emerging gauge
field dynamics. We discuss also properties of the lowest energy competing
orders.
