# Understanding the Monster CFT

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I try to understand what the Monster CFT and its possible connection to 3 dimensional gravity at ($c=24$) is about (see https://arxiv.org/abs/0706.3359)

To my best understanding (and please correct me if here are anywhere wrong statements) the Monster CFT has an extended (with respect to the Virasoro algebra) chiral algebra.

1. Do the elements of the extended chiral algebra, other than the Virasoro algebra, also create new states (like e.g., models with affine Kac-Moody algebras)?

The Virasoro primary fields fall into irreducible representations of the Monster.

1. How many irreducible representations are there? Are there finitely many of them?

At least according to http://www.ams.org/notices/200209/what-is.pdf there are 194 complex irreducible representations.

If I understand this correctly, then the coefficients of the $J$ invariant (see https://en.wikipedia.org/wiki/Monstrous_moonshine , $r_n$ is the dimension of the irreducible representation $r_n$)

$J(q)=r_1 q^{-1}+ (r_1+r_2) q+ (r_1+ r_2+r_3) q^2+ ...$

should get at some point no new contributions from new representations, i.e., there should be no term $r_{195}$.

1. Did I understand this correctly?

If that is correct then I'm puzzled by the following: The dimension $r_n$ means that there are $r_n$ new Virasoro primary fields at that this level. These in turn should "correspond" to black holes (https://arxiv.org/abs/0706.3359).

1. If 3. is correct, why should there be no new Virasoro primaries after some level. What happens to the black holes at this level?

This post imported from StackExchange Physics at 2017-08-26 13:34 (UTC), posted by SE-user ungerade

edited Aug 26, 2017
IIRC, the Monster CFT is studied in math under the guise of "Monstrous Moonshine" (I've seen it called the "Moonshine Module" in the math literature). A great introductory book on this is Gannon's Moonshine Beyond the Monster

This post imported from StackExchange Physics at 2017-08-26 13:34 (UTC), posted by SE-user Alex Nelson
(And all finite groups have a finite number of irreducible representations, equal to the number of conjugacy classes in the group in fact.)

This post imported from StackExchange Physics at 2017-08-26 13:34 (UTC), posted by SE-user Alex Nelson
@AlexNelson : That seems to partially answer 3. and makes 4. more likely to be a real question ;). There is a lot of literature about the Monster CFT but for most of it it would be quite time consuming (at least for me) to study it to thoroughly. My hope is that somebody knowledgeable enough can answer the questions.

This post imported from StackExchange Physics at 2017-08-26 13:34 (UTC), posted by SE-user ungerade
It seems like references [24,25] in the linked paper are good places to start answering Q4. "The dimension of a given irrep for the Monster is the number of primary fields at this level" seems to be covered in those references, IIRC. How these "correspond" to black holes, Witten seems to use the primary state $\mid\Lambda\rangle$ corresponding to a primary operator, as discussed on the bottom of page 32 (p33 of the pdf).

This post imported from StackExchange Physics at 2017-08-26 13:34 (UTC), posted by SE-user Alex Nelson
@AlexNelson The question is, if there are only finitely many irreducible representations then at some level there are no new primary states any more. That would imply that there are no black holes (or at least none with entropy) for that level (and mass). Given that the BTZ black holes have no upper bound on the mass I do not understand why this should be the case.

This post imported from StackExchange Physics at 2017-08-26 13:34 (UTC), posted by SE-user ungerade

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The formula written for the first three terms of J(q) is correct but it could suggest that in general the coefficient of $q^{n+1}$ is $r_1 + ...+r_n$. But if one checks the wikipedia page, one sees that it is not correct.

The monster group has 194 irreducible representations. At every level n, we have a representation of the monster group and so a decomposition in irreducible representations $a_{1,n}r_1+\dots+a_{194,n}r_{194}$. It is not true that new primary fields at level $n$ have to be in the representation $r_n$, they can be in any of the representations $r_1, \dots, r_{194}$.

Remark: a 2d CFT with finitely many primary fields (with respect to theVirasoro algebra) has necessary c<1. The fact that the monster CFT has c=24 is another reason why it has infinitely primary fields.

answered Aug 27, 2017 by (5,140 points)
edited Aug 27, 2017 by 40227

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