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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?

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Take a look at mathoverflow.net/questions/13851/… for information on realizing the Monster group as a Galois group over Q. No mention there of applications to physics, so this is not a direct answer to the question. –  KConrad Mar 20 '11 at 19:52
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There is a book called "Moonshine Beyond the Monster: the bridge connecting Algebra, Modular Forms and Physics" which was written by Terry Gannon. I think it's probably a good idea for you to take a look at it ;). –  Max Muller Mar 21 '11 at 8:11
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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.

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The wikipedea link for the "no ghost" theorem is en.wikipedia.org/wiki/Goddard%E2%80%93Thorn_theorem –  Gil Kalai Mar 21 '11 at 19:30
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I know very little about the subject; But you can check this lecture : cosmolearning.com/courses/… by Ed Witten where he links the Monster group to quantizing gravity in 2+1 dimensions. In particular, the lecture contains a very interesting derivation of black hole entropy in 2+1 as $log(196883) = 12.19$.. , where the classical entropy is $4*\pi = 12.57$ which is off by a few percent. What do you think of that professor Borcherds? –  Mohamed Alaa El Behairy Jun 9 '11 at 6:28
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I know very little about the subject; But you can check this lecture : cosmolearning.com/courses/… by Ed Witten where he links the Monster group to quantizing gravity in 2+1 dimensions. In particular, the lecture contains a very interesting derivation of black hole entropy in 2+1 as log(196883)=12.19.. , where the classical entropy is 4∗π=12.57 which is off by a few percent. What do you think of that professor Borcherds?

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Witten's work in this area is well-known, and he has written a preprint that was widely read: arxiv.org/abs/0706.3359 This does not seem to be an application of the monster group in physics, but it indicates the potential for a relationship to exist. –  S. Carnahan Jun 10 '11 at 4:54
    
Thanks for the suggestion; But can you elaborate more why this does not seem to be an application of the monster group in physics ? –  Mohamed Alaa El Behairy Jun 10 '11 at 5:32
    
Witten doesn't use the monster group anywhere in his paper. He describes a conjectural correspondence between pure 3d gravity in $AdS_3$ space and extremal conformal field theories. In the case of central charge 24, the only known extremal CFT has monster symmetry, but the uniqueness of the moonshine module as an extremal CFT with that central charge is still conjectural, and that symmetry does not seem to appear naturally on the gravity side. In particular, the entropy calculation uses a 196883 dimensional vector space, but does not use a monster action on it. –  S. Carnahan Jun 11 '11 at 13:58
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