What are the $f$-modes, $g$-modes,...of GW in neutron stars and compact objects? Reading gravitational wave astronomy papers, sometimes is mentioned the importance and relevance of $f$-modes and $g$-modes of gravitational waves.
What are exactly the $f$-modes/$g$-modes in gravitational waves and what are their physical meaning?
In advance, in the answer could be included if are they related or could they be related to quantum excitations $\ell=0,1,2,3,...$($s,p,d,f,g,h,...$) modes?
 A: Take a look at Asteroseismology, the modes they are referring to are related to the modes of oscillation within the star. In particular Neutron star Oscillations  that have a wider variety of modes. You will also see people talk about fluid oscillations in the context of stars and in some sense the modes are the way the fluid vibrates, or resonates and the star deforms slightly.
p-modes are pressure modes or acoustic like modes
g-modes are called gravity modes, due to buoyancy forces within the stat (but are not gravitational waves)
f-modes are the fundamental modes I think observed on the surface of the star.
I think the reason you are seeing them discussed in gravitational wave papers is that the gravitational ways are detected when black holes of neutron stars interact. I suppose that when the neutron star is disturbed by the other star spiraling in that resulting gravitational wave that is generated will have its spectrum modified by the mode structure of the star since the energy will preferentially go into those modes.
The modes are generally related to spherical harmonics. So in some sense the math is similar to what you see in quantum mechanics.
