# Understanding the Monster CFT

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?
• 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 – Alex Nelson Jun 8 '17 at 15:58
• (And all finite groups have a finite number of irreducible representations, equal to the number of conjugacy classes in the group in fact.) – Alex Nelson Jun 8 '17 at 17:32
• @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. – ungerade Jun 8 '17 at 19:50
• 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). – Alex Nelson Jun 9 '17 at 16:16
• There is an answer at physicsoverflow.org/39539 – Arnold Neumaier Sep 11 '17 at 13:15