I am doing a project on the inverse Galois problem, and am seeking to show that the monster group is realisable over the rationals. I have heard that the monster group has found uses in theoretical physics, and was wondering what those uses might be. Also, is there any practical significance to theoretical physics in the result I am aiming to prove?
There are presently no applications of the monster group in physics, though there is a lot of misleading speculation about this. However in the other direction there are some applications of ideas from physics to the monster group. In particular the no-ghost theorem in string theory is used to construct the monster Lie algebra acted on by the monster group.
The ideas from physics seem to have no direct connection with the problem of realizing the monster as a Galois group over the rationals. This was solved by Thompson, who showed by character-table calculations that the monster satisfies a sufficient condition ("rigidity") for it to be a Galois group.