  We study strongly correlated ground states of dipolar fermions in a honeycomb
optical lattice with spatial variations in hopping amplitudes. Similar to a
strained graphene, such nonuniform hopping amplitudes produce valley-dependent
pseudomagnetic fields for fermions near the two Dirac points, resulting in the
formation of Landau levels. The dipole moments polarized perpendicular to the
honeycomb plane yield a long-range repulsive interaction. By exact
diagonalization in the zeroth-Landau-level basis, we show that this repulsive
interaction stabilizes a variety of valley-polarized fractional quantum Hall
states such as Laughlin and composite-fermion states. The present system thus
offers an intriguing platform for emulating fractional quantum Hall physics in
a static optical lattice. We calculate the energy gaps above these
incompressible states, and discuss the temperature scales required for their
experimental realization.
