  I study cross dimensionality of $p$-orbital atomic fermions loaded in an
optical square lattice with repulsive interactions. The cross-dimensionality
emerges when the transverse tunneling of $p$-orbital fermions is negligible.
With renormalization group analysis, the system is found to support two
dimensional charge, orbital, and spin density wave states with incommensurate
wavevectors. The transition temperatures of these states are controlled by
perturbations near a one dimensional Luttinger liquid fixed point. Considering
transverse tunneling, the cross-dimensionality breaks down and the density wave
(DW) orders become unstable, and I find an unconventional superconducting state
mediated by fluctuation effects. The superconducting gap has an emergent nodal
structure determined by the Fermi momentum, which is tunable by controlling
atomic density. Taking an effective description of the superconducting state,
it is shown that the nodal structure of Cooper pairing can be extracted from
momentum-space radio-frequency spectroscopy in atomic experiments. These
results imply that $p$-orbital fermions could enrich the possibilities of
studying correlated physics in optical lattice quantum emulators beyond the
single-band Fermi Hubbard model.
