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I was puzzling if there were anything trying to explain $M_p^2=NM(higgs)^2$ or $M_p^2=NM(top)^2$ and I found it that $N\sim N(BM)$, where $N(BM)$ is the order of the Baby Monster group, about $\sim 4\cdot 10^{33}$. My question is: what kind of lattice model describe the Baby Monster and if that could be related somehow to some compatification in any known string theory, just as the Monster group arises in 26d, when reducing it to 24, and use the Leech lattice in bosonic string theory. *Extra: are there other moonshines for the baby monster and other sporadic groups being used in string theory or superstrings?

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First, a reminder of how monstrous moonshine fits into string theory, as I understand it: The Monster is associated with the 24-dimensional Leech lattice; you can compactify the 26-dimensional bosonic string on the Leech lattice; Witten proposes that the worldsheet CFT of that compactified string is dual to pure gravity in AdS3 for a particular negative cosmological constant.

The counterpart of the Leech lattice for the Baby Monster is the 23-dimensional "shorter Leech lattice". There is a Baby Monster CFT constructed by Gerald Höhn, whose automorphism group is Z_2 x the Baby Monster. Presumably one can compactify the bosonic string on the shorter Leech lattice too (leaving 2+1 uncompactified dimensions), but I don't know if Höhn's CFT is the worldsheet CFT of strings in such a compactification.

There are also many papers on "umbral moonshine" in string theory. "Pariah moonshine" hasn't made it into mathematical physics yet, although Miranda Cheng (a guru of umbral moonshine) has at least one purely mathematical paper citing it.

None of that has any known relationship to phenomenology, and I do not see how the order of the group can translate into a mass ratio. There is actually a paper on arxiv claiming to obtain the Higgs mass from the cube root of the order of the Monster group, but the argument didn't make sense to me (I analyzed some of it here).

You might also be interested to know that a "shorter Leech liquid" has been described, as a hypothetical phase of condensed matter.

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