One way to define the Monster group is to consider a conformal field theory (CFT) corresponding to central charge $c=24$ and look at the automorphism group of its vertex operator algebra. For one of the CFT's with $c=24$ this is indeed the Monster group.

What would be the corresponding group for CFT's with $c \neq 24$?

Is there a condition on the CFT or on the central charge such that the automorphism group of the vertex operator algebra is a finite non-trivial group? Or some condition such that it is a finite dimensional Lie group?

For the minimal models with $c = 1 - \frac{6}{m(m+1)}$ it is a trivial group. Do we have examples for well-known or not so well-known CFT's where the automorphism group of its vertex operator algebra is non-trivial and known (apart from $c=24$ and the Monster)?

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    $\begingroup$ There is not a unique $c=24$ CFT, not even a unique $c=24$ holomorphic CFT. Conjecturally there are $71$ of them and only one has Monster symmetry. Your question is way too vague as there is no notion of ${\it the}$ automorphism group of a VOA that depends only on $c$. $\endgroup$ – Jeff Harvey Apr 24 '15 at 15:27
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    $\begingroup$ For the minimal models it should be the trivial group. $\endgroup$ – Marcel Bischoff Apr 26 '15 at 21:56
  • $\begingroup$ I think that the weight 1 space has to be trivial, in order for the automorphism group to be discrete. Note there is something else special about the Moonshine VOA, namely it is self-dual, i.e. the only simple module is the VOA itself. This implies that $c \mod 8 =0$. $\endgroup$ – Marcel Bischoff Apr 27 '15 at 14:16
  • $\begingroup$ Thanks, that's a very useful piece of information! Basically I'm after some intuition about the Monster group; what other sorts of group appear for other CFT's as automorphisms of its VOA and what is so special about that particular $c=24$ CFT. $\endgroup$ – DanielFetchinson Apr 27 '15 at 18:25

As others have mentioned, there are many CFTs, but we can narrow down our list by looking at conditions that select for interesting automorphism groups. Perhaps the easiest is to consider holomorphic ($C_2$-cofinite) vertex operator algebras. By Zhu's theorem, these necessarily have central charge given by a nonnegative multiple of 8, so we can list them in some rough increasing order.

$c=0$: This is just the field of complex numbers, with trivial symmetry (as a ring over itself).

$c=8$: There is one example: the $E_8$ lattice algebra. Its automorphism group is the Lie group $E_8(\mathbf{C})$.

$c=16$: There are two cases, given by the lattice algebras for $E_8 \times E_8$ and $D_{16}^+$, and the automorphism groups are $E_8 \wr 2$ and $SO^+(32)$. This case and the previous case were proved by Dong and Mason.

$c=24$: In 1992, Schellekens conjectured that these are classified by their weight 1 Lie algebras, and gave 71 candidate types. The last candidate vertex operator algebra was finally constructed 6 weeks ago by Lam and Lin. Nontrivial Lie algebras yield positive-dimensional Lie groups of symmetries, so the only discrete case is where the Lie algebra is zero. The uniqueness question is still open, but if it is true, then the Monster is the only discrete automorphism group. If there is a counterexample to uniqueness of the moonshine module, then we still have a lot of control of its automorphisms due to genus zero constraints. That is, most of Borcherds's argument still works for such an object.

$c \geq 32$: The number of isomorphism types grows very quickly, just because they contain the vertex operator algebras attached to positive definite even unimodular lattices. For example, with $c=32$, King's mass formula gives over $10^9$ lattices, and over $10^7$ lattices with no roots. It is quite likely that by orbifolding some of the latter types, we get discrete symmetry groups, but it seems no one has bothered to work out any cases completely.

What we see from this list is that, from the lens of the category of holomorphic vertex operator algebras, the first nontrivial group is $E_8$, and the first nontrivial finite group is the Monster. At large central charge, we expect things to become boring from a symmetry standpoint, in much the same way that high rank lattices and high genus curves tend to become boring.

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One possible answer to your question is discussed in Witten's paper Three-dimensional gravity revisited.

Since the beginning of the subject, it has been conjectured that the Moonshine VOA is the unique self-dual VOA with $c=24$ and no spin=1 fields. As Marcel mentions in the comments, the spin=1 fields comprise the Lie algebra of the automorphism group, so having no spin=1 fields means that your automorphism group is discrete. Any $c=24$ self-dual VOA with no spin=1 fields necessarily has nontrivial spin=2 fields. This is because the torus partition function is a modular function of shape $q^{-1} + 0 + O(q)$, the $0$ counting the number of spin=1 fields. There is a unique modular function of that shape, namely $j(q) - 744 = q^{-1} + 0 + 196844q + \dots$. So there must be 196844 spin=2 fields, of which one is the energy-momentum tensor (conformal vector) and the other 196843 are Virasoro primaries.

Similarly, suppose you have a $c=24k$ self-dual VOA. Then its partition function is a modular function beginning $q^{-k} + \dots$. The descendants of the vacuum contribute $q^{-k} \prod_{n\geq 2} (1 - q^n)^{-1}$ to the torus partition function, and the remaining Virasoro primaries lead to other contributions. Hohn calls a self-dual VOA extremal if it has "as few Virasoro primaries as is possible". More precisely, there is a unique modular function of shape $q^{-k} \prod_{n\geq 2} (1 - q^n)^{-1} + O(q)$; a VOA is extremal if its torus partition function is this one. Extremal VOAs are higher-$c$ generalizations of the Moonshine VOA.

Witten conjectures that there is a unique extremal CFT for each $k$. Specifically, he conjectures that it is the holographic dual to 3d quantum gravity with negative cosmological constant and $k = $ some particular dimensionless combination of the paramaters (cosmological constant, gravitational constant, etc.) in the gravity model. He suggests that the other Virasoro primaries, starting in spin=$(k+1)$, correspond to black hole states in the bulk. He bases his conjecture on numerical agreement in the large-$k$ limit between the number of low-lying black hole states and the entropy of a classical black hole. (The agreement is good even at small $k$.)

In spite of the conjecture, other than the Moonshine VOA there are no known extremal VOAs.

(Witten also formulates a notion of extremality for $\mathcal N=1$ SVOAs, in which case $c = 12k^*$; one can pose similar conjectures. The known-to-be-unique $k^*=1$ extremal SVOA is Duncan's "Conway Moonshine" SVOA (I believe it belongs to Duncan, although it is much older, appearing in the FLM work and in Borcherds' work on positive-characteristic VOAs). A $k^*=2$ extremal SVOA is known; it is an odd analog of the Moonshine VOA. Beyond this the story is like in the bosonic case: extremal SVOAs are not known to exist.)

Witten shows that the coefficients in the $q$-expansion of the modular function of shape $q^{-k} \prod_{n\geq 2} (1 - q^n)^{-1} + O(q)$ are the dimensions of Monster representations. Based on this (and some further evidence), he conjectured that the Monster acts on all extremal VOAs (and is the symmetry group of 3d gravity.) Gaiotto showed that the conjecture fails in that if there is an extremal VOA with $k=2$, then it does not have Monster symmetry. I believe almost nothing is known about the symmetry groups of the not-known-to-exist extremal VOAs.

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  • $\begingroup$ All your products involve $(1+q^n)^{-1}$ factors. Is that what you really mean, or should they have been $(1-q^n)^{-1}$? The latter makes more sense in a bosonic theory. $\endgroup$ – Victor Protsak Feb 5 '17 at 0:55
  • $\begingroup$ @VictorProtsak Yes, sorry, you are correct. $\endgroup$ – Theo Johnson-Freyd Feb 5 '17 at 4:59
  • $\begingroup$ I think I fixed the signs. $\endgroup$ – Theo Johnson-Freyd Feb 5 '17 at 5:00
  • $\begingroup$ Minor correction: any VOA has spin=2 fields, since the conformal vector is one $\endgroup$ – W. Cadegan-Schlieper Feb 6 '17 at 3:46
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    $\begingroup$ It is probably worth pointing out that Maloney-Witten (2007) have argued against the existence of a pure 3d gravity theory, due to some failed consistency checks. Pure chiral gravity appears to be more favored as a candidate dual to extremal theories. Also, there are some claims that the extremality condition can be relaxed somewhat without destroying the duality. Finally, Gerald Höhn independently showed the nonexistence of monster symmetry in the $k=2$ case. $\endgroup$ – S. Carnahan Feb 7 '17 at 8:53

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