What is the definition of light-cone gauge? Is there any advantage for working in this gauge?
Are you referring to string theory in particular (if so you might want to post in the Beyond The Standard Model subforum)? If yes, you should look at the book by Zwiebach where it is explained in details. Basically, it's the gauge where light-cone coordinates are used for the coordinates of string. The advantage in that gauge is that it's a "physical gauge" in the sense that only physical degrees of freedome are present from the very beginning. It's a bit like quantizing QED with only transverse polarizations of the photon present as the degrees of freedom. The disadvantage (like in QED) is that the theory quantized in that gauge is not manifestly Lorentz invariant and one has to check this carefully.
** the book by Zwiebach where it is explained in details
I can confirm that a detailed description of the light cone gauge is in Barton Zwiebach's book "A first course in string theory". It is part of general relativity and not part of an extension beyond the standard model.
To my knowledge, string theory started from strong force theory, so I think it will be better
answer the question from QCD. Maybe this question is nothing to do with string theory.
In QCD, light-cone gauge is defined as A^+=0. It is reasonable as only one direction of the incident particle is comparable to light speed in high energy experiments. It is much clearer
from the Lorentz transformation matrix, the boost is only involved by one space axes and one time axes.