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Frenkel, Lepowsky, Meurman constructed the vertex operator algebra (VOA) $V^\natural$ as a chiral orbifold CFT whose target space is $\mathbb{R}^{24}/\Lambda/\mathbb{Z}_2$. (Here the last $\mathbb{Z}_2$ is a group of order two which acts by sending $\vec x\to -\vec x$, and not the ring of 2-adic integers. Sorry for using physics notation.)

This construction makes (almost) manifest that the said VOA has $2^{1+24}_{+}\cdot Co1$ as part of its symmetry group. In addition, there is an operation $\sigma$ which mixes the twisted sector and the untwisted sector. Then, adding $\sigma$ to $2^{1+24}_+\cdot Co1$ gives rise to the monster simple group $M$. Then $M$ has another involution $z$ (possibly conjugate to $\sigma$) such that $2^{1+24}_+\cdot Co1$ is the centralizer of $z$, $C_G(z)$. (Honestly I don't really understand the construction, but that's the story I've heard.)

Now, there are many other simple groups $G$ which has a similar structure, i.e. there is an involution $z\in G$ such that the centralizer $C_G(z)$ has the structure $2^{1+n}.H$.

Then my question is: is it always the case that there is a lattice $L$ of rank $n$ whose symmetry is $H$ (and $-1$) such that the VOA based on $\mathbb{R}^{n}/L/\mathbb{Z}_2$ has the symmetry $G$, given by adjoining an operation $\sigma$ mixing the twisted and the untwisted sector, to the part $2^{1+n}.H$ which exists almost by construction?


I realized now that Jeff Harvey already asked almost the same question a few months ago, see this MO question. I even made a comment in that thread; I completely forgot about that. You see, I'm only slowly digesting the interesting problem Jeff raised...

But of course the moderators can close my question as an exact duplicate. Sorry about that.

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I think you should replace the words "exchanges" and "exchanging" with "mixes" and "mixing", since an automorphism will not send the vacuum to anything but the vacuum. Also, the element $z$ you mention should be distinct from $\sigma$, although it may be conjugate. – S. Carnahan Mar 2 '11 at 15:01
Thanks for the corrections. – Yuji Tachikawa Mar 3 '11 at 2:19

I have not studied FLM in complete detail, but don't think the involution $\sigma$ is sufficiently natural to make the generation of $G$ an automatic process. That is to say, even if it does always exist as an automorphism of the VOA, it may be difficult to find without explicit combinatorial information about the lattice.

Perhaps the best place to look for similar constructions is John Duncan's paper on the Conway group, where he gives an explicit construction of a $N=1$ vertex superalgebra with a sporadic automorphism group.

Slightly off-topic: There is an alternative construction of the monster VOA as a $\mathbb{Z}/3$ orbifold outlined in a 1994 conference proceedings paper by Dong and Mason (The construction of the moonshine module as a $\mathbb{Z}/p$-orbifold), and there is an analysis of the orbifold theory with no claims about the automorphism group given in a paper by Montague. The Dong-Mason paper promises the full details of proving that the automorphism group is in fact the monster simple group in a forthcoming paper that has yet to appear. I believe it also uses an involution but the calculations are apparently even more complicated than the order 2 case.

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Thanks for the info. Before talking about the involution $\sigma$, is there a lattice $L$ of rank $n$ for each such $H$ for a simple group $G$? – Yuji Tachikawa Mar 3 '11 at 4:39
That is a good question whose answer I do not know. – S. Carnahan Mar 3 '11 at 8:55

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