  We develop a theory of weakly interacting fermionic atoms in shaken optical
lattices based on the orbital mixing in the presence of time-periodic
modulations. Specifically, we focus on fermionic atoms in circularly shaken
square lattice with near resonance frequencies, i.e., tuned close to the energy
separation between $s$-band and the $p$-bands. First, we derive a
time-independent four-band effective Hamiltonian in the non-interacting limit.
Diagonalization of the effective Hamiltonian yields a quasi-energy spectrum
consistent with the full numerical Floquet solution that includes all higher
bands. In particular, we find that the hybridized $s$-band develops multiple
minima and therefore non-trivial Fermi surfaces at different fillings. We then
obtain the effective interactions for atoms in the hybridized $s$-band
analytically and show that they acquire momentum dependence on the Fermi
surface even though the bare interaction is contact-like. We apply the theory
to find the phase diagram of fermions with weak attractive interactions and
demonstrate that the pairing symmetry is $s+d$-wave. Our theory is valid for a
range of shaking frequencies near resonance, and it can be generalized to other
phases of interacting fermions in shaken lattices.
