Orbital Symmetry and Orbital Excitations in High-$T_c$ Superconductors

We discuss a few possibilities of high-$T_c$ superconductivity with more than one orbital symmetry contributing to the pairing. First, we show that the high energies of orbital excitations in various cuprates suggest a simplified model with a single orbital of $x^2-y^2$ symmetry doped by holes. Next, several routes towards involving both $e_g$ orbital symmetries for doped holes are discussed: (i) some give superconductivity in a CuO${}_2$ monolayer on Bi2212 superconductors, Sr${}_2$CuO${}_{4-\delta }$, Ba${}_2$CuO${}_{4-\delta }$, while (ii) others as nickelate heterostructures or Eu${}_{2-x}$Sr${}_x$NiO${}_4$, could in principle realize it as well. At low electron filling of Ru ions, spin-orbital entangled states of $t_{2g}$ symmetry contribute in Sr${}_2$RuO${}_4$. Finally, electrons with both $t_{2g}$ and $e_g$ orbital symmetries contribute to the superconducting properties and nematicity of Fe-based superconductors, pnictides or FeSe. Some of them provide examples of orbital-selective Cooper pairing.