My question concerns applications of monstrous moonshine, which is the connection between the $j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, to a possible form of 3D quantum gravity, as seen in the article below:


However, my question is what are other applications of monstrous moonshine to physics outside string theory? Are there possible connections to conformal field theories in condensed matter physics? If not to condensed matter, what are some other possible physical applications?

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    $\begingroup$ So far you are none for zero because string theory is not even false, yet. In other words, it is not canonical physics by any means. It is not clear to me how or why one would even want to apply a discrete group of such size to a condensed matter application which does not have more than a handful of approximate symmetries, at most (and truthfully, most condensed matter physics problems of interest are actually about the question of what happens when the symmetry breaks on dislocations etc.). $\endgroup$ – CuriousOne Jun 3 '15 at 6:48

This is a very good question and one that is quite difficult to answer. In physics, the use of a group is usually to describe a symmetry of the physical system. What we know of the monster is that it has a non-trivial action in 196883 dimensions. In other words, the Monster group is the symmetry of a 196883 dimensional object. The hope of Monster moonshine for a string theorist was that it would highlight some of the symmetries of string theory, which we know very little about.

One application that immediately jumps to mind is in coding theory and error correction and many other moonshines (Conway group, Mathieu group) could shine a light on this application but there is some inspiration from the Monster moonshine.

Let's look at the Leech lattice, $\Lambda_{24}^L$. It is a 24 dimensional, unimodular, even self dual lattice (whose automorphism group happens to be $Co_0$, another sporadic group). When one takes the CFT of 24 free bosons compactified on the Leech lattice with a further $\mathbb Z_2$ orbifold (which was actually I believe the first construction of the asymmetric orbifold), the resulting vertex operator algebra, known as the Griess algebra has an automorphism group which is the Monster. The partition function of this VOA is the Klein $j-$ function (minus a constant). The Leech lattice however has applications to things outside of the string theoretic settings. In coding theory/information theory. The Golay code is of focus here. it is a binary code of 24 letters which is capable of correcting three errors. The extended version of this Golay code which is a 24 dimensional code is can be embedded naturally into the Leech lattice, for which we already know some group theoretic properties. This Golay code in fact appears in the Ramond-Ramond ground state of $\mathcal N = 4$ string theory with $K3$ target and is therefore related to Moonshine (Mathieu Moonshine). (Harvey, et.al, Harvey, Moore (2020)) One really interesting use of the Golay code was in the flight of the Voyager.

These error correcting codes can also be generalized to representations of superalgebras. One key aspect of this is to construct quantum error correcting codes whose representations sit inside a lattice, whose automorphism group is a sporadic group. In this way, moonshine has some application outside of string theory. There were many emminenent mathematicians who thought that sporadic groups were not as interesting and that there were more challenging problems, and at the same time others who did work on finite simple groups who believed it to be an essential problem with no applications in/insight from physics. Both groups of mathematicians are/were right and wrong to a certain extent.


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