The high point of Friday’s string cosmology workshop here in New York was Witten’s lecture on his new ideas about 2+1 dimensional quantum gravity. I’ll try and reproduce here what I understood from the lecture, but this (2+1 d quantum gravity) is not a subject I’ve ever followed closely, so my understanding of the topic is very limited. It does seem clear to me though that Witten has come up with a striking new idea about this subject, linking together some very beautiful mathematics and physics. He has yet to write a paper on the subject, but presumably there will be one appearing relatively soon. I also suspect this is what he’ll be talking about at Strings 2007.

Witten began by stating his motivation: to study fully quantum black holes in an exactly solvable toy model. There’s no exactly solvable model in 3+1d, and 1+1d is too simple, so that leaves 2+1d. Assuming 2+1d, for positive cosmological constant Λ he is suspicious that the theory is non-perturbatively unstable and one can’t get precise observables, for Λ=0 one doesn’t have black holes, so that leaves negative Λ, here the vacuum solution is anti-deSitter space, AdS_{3}.

Quantum gravity in AdS_{3} is related to 2d conformal field theory. There have been studies of AdS_{3}/CFT_{2} as a lower dimensional version of string/gauge duality, but here he uses not string theory on AdS_{3}, but a quantum field theory. In a question afterwards, someone asked about string theory, and Witten just noted that perhaps what he had to say could be embedded in string theory, and that the recent Green et. al. paper showing that one can’t get pure supergravity by taking a limit of string theory did not apply in 3d. If one wants to interpret this new work in light of the the LQG/string theory wars, it’s worth noting that the technique used here, reexpressing gravity in terms of gauge theory variables and hoping to quantize in these variables instead of using strings, is one of the central ideas in the LQG program for quantizing 3+1d gravity. Witten was careful to point out though that there was no 3+1d analog of what he was doing, claiming that one can’t covariantly express gravity in terms of gauge theory in 3+1d (he said that LQG does this non-covariantly).

For negative Λ the theory has so-called BTZ black hole solutions, discovered by Banados, Teitelboim and Zanelli back in 1992, and it is for the quantum theory of these black holes that Witten is trying to find an exact solution. The technique he uses is one that goes back to the 80s, that of re-expressing the theory in terms of SO(2,1) (or its double cover SL(2,**R**)) gauge theory, where the action becomes the Chern-Simons action. More precisely, the Einstein-Hilbert action

$$I_{EH}=\frac{1}{16\pi G}\int d^3x\sqrt{g}(R +2/l^2)$$

(here the cosmological constant is $\Lambda=-1/l^2$) gets rewritten as an SO(2,2)=SO(2,1)XSO(2,1) gauge theory with connection

$$A= \begin{pmatrix}\omega & e \\ -e & 0 \end{pmatrix}$$

where &\omega; is a 3X3 matrix (the spin-connection), e is the 3d vielbein, and the gauge theory action is the Chern-Simons action

$$I=\frac{k^\prime}{4\pi}\int Tr(A\wedge dA+\frac{2}{3}A\wedge A\wedge A)$$

with $k^\prime=\frac{l}{4G}$ (that 4 may not be quite right…).

Witten wants to exploit the relation between this kind of topological QFT and 2d conformal field theory that he first investigated in several contexts (including one that won him a Fields medal) back in the late eighties. He notes that in this context the existence of left and right Virasoro symmetries with central charges $c_L=c_R=\frac{3l}{2G}$ was first discovered by Brown and Henneaux back in 1986, and he refers to this discovery as the first evidence of an AdS/CFT correspondence. If one really does have a CFT description, one expects that the central charges can’t vary continuously, but that 2+1d gravity will only make sense for certain values of $l/G$, but Witten notes that there is no rigorous way to find the right values one will get upon quantization.

He then goes on to make a “guess”, adding to the action a multiple of the Chern-Simons invariant of the spin connection

$$I^\prime=\frac{k}{4\pi}\int Tr(\omega\wedge d\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega)$$

Now the theory depends on two parameters: $l/G$ and an integer k.

Using the fact that SO(2,2)=SO(2,1)XSO(2,1), one can rewrite the total action as the sum of two Chern-Simons terms

$$I= \frac{k_L}{4\pi} \int Tr(A_-\wedge dA_-+\frac{2}{3}A_-\wedge A_-\wedge A_-)$$

$$ \ \ + \frac{k_R}{4\pi}\int Tr(A_+\wedge dA_++\frac{2}{3}A_+\wedge A_+\wedge A_+)$$

for connections

$$A_{\pm}=\omega\pm e$$

Now instead of $l/G$ and k we have $k_L,k_R$ and these are quantized if we take the gauge theory seriously. By matching Chern-Simons and gravity the central charges turn out to be

$$ (c_L, c_R)= (24k_L, 24k_R)$$

and holomorphic factorization is possible in the 2d CFT for just these values

Looking at just the holomorphic part, we have a holomorphic CFT with central charge c=24k and ground state energy -c/24=-k (note, now a different k than before…).

The partition function is expected to be ($q=e^{-\beta}$)

$$Z(q)=q^{-k}\Pi_{n=2}^\infty \frac{1}{1-q^n}$$

The first term in the product is the ground state (AdS_{3}), the only primary state, with the other terms Virasoro descendants (excitations of the vacuum from acting with the stress-energy tensor and derivatives).

Witten then goes on to note that this expression is not modular invariant, so one expects other terms in the product, corresponding to other primary states. By an argument I didn’t understand he claimed that these would be of order $q^{1}$, at an energy k+1 above the ground state, and his proposal was that it would be this modular invariant function that would include black hole states.

In these units the minimum black hole mass is M=k, but here one is getting states only at mass M=k+1 and above. This is because the Bekenstein-Hawking entropy of the M=k black hole is 0, so it doesn’t contribute to the partition function.

Witten claimed that this proposal gives degeneracies of states that agree with the Bekenstein-Hawking entropy formula. As an example, for k=1 the partition function is given by the famous J-function

$$J(q)=j(q)-744=q^{-1}+196884q+\ldots$$

and thus for a black hole of mass 2 the number of primaries is 196883 and the entropy is ln(196883)=12.19, which can be compared to the Bekenstein-Hawking semi-classical prediction of 12.57 (one only expects agreement for large k,M).

The number 196883 is famous as the lowest dimension of an irreducible representation of the monster group, and this partition function is famous as having coefficients that give the dimensions of the other irreducibles (“modular moonshine”). There is a conjecture that there is a unique CFT with this partition function. If so, it must be the CFT that has the monster group as automorphism group. It has always seemed odd that this very special CFT didn’t correspond to a particularly special physical system, but if Witten is right, now it has an interpretation in terms of the quantum theory of black holes in 2+1 dimensions.

Anyway, that’s what I was able to understand of what Witten had to say and what he was claiming. Other people have worked on this problem in the past, for a recent review article on this topic by Carlip, see here. Carlip describes the understanding of the problem at the time as “highly incomplete”, and one of the explanations he describes relates the black hole problem to the Liouville theory. A question from the audience after the talk asked about this, and Witten indicated that he thought the Liouville theory explanation did not work.

I’m no expert here, so unclear on the details, why some of these things might be true, and what the implications might be, but this does seem to be a remarkable new idea, involves beautiful mathematics, and seems to provide promising insight into a crucial lower dimensional toy model. I suspect it will draw a lot of attention from theorists in the future.

For this posting, I especially encourage any comments from people more knowledgable than myself who can correct anything I’ve got wrong. I also strongly discourage people who know little about this from contributing comments that will add noise and incorrect information. Bad enough that I’m trying to provide information about something I’m not expert on; if you can help that’s great, but if not, please don’t make it worse…

**Update**: Lubos has picked up on this, which he describes as having been “leaked”, and gives the usual argument that this must be part of string theory.

Assuming 2+1d, for positive cosmological constant Λ he is suspicious that the theory is non-perturbatively unstable and one can’t get precise observables,for Λ=0 one doesn’t have black holes, so that leaves negative Λ, here the vacuum solution is anti-deSitter space, AdS3.Stupid question: Why not?

What happens if you just take a certain amount of mass and drop it into an area of zero-cosmological-constant 2+1d space less than that mass’s Schwarzschild radius, or whatever?

I just deleted almost all the comments on this posting, since, besides one helpful comment pointing out a typo, they were doing what I specifically requested people not to do. Please try and resist the temptation to add to the noise level here. If you have an informative and substantive comment, please post it, but the level of not-worth-reading comments followed by people pointlessly bickering about them has gotten really annoying. This creates an environment I wouldn’t want to participate in myself.

Coin,

I don’t know what the physical interpretation is, but I assume the point is that the BTZ solutions don’t exist for zero CC. Maybe someone more familiar than me with this has insight into why this is.

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2+1 is special: For lambda = 0, Ricci = 0 implies Riemann = 0; there’s no Birkhoff theroem in 2+1. So, a vaccum equation in 2+1 is just flat space. No curvature, so no black holes.

More intuitively, in 3+1 one can consider the lower bound of a tottally collapsed gravitational object (Schwarzschild radius = 0) as a critical point between wave dispersion and gravitational / self attraction. Consider a radially ingoing gaussian profile of energy; say some field. For small amplitude the dispersion will exceed the self attraction and the solution puts the field on the boundary at future infinity. With increasing amplitude, eventually one finds the critical point and black holes thereafter.

In 2+1, for Lambda >=0 there is not the same mechanism of self attraction; so no black holes. It’s all dispersive instead.

woit/anon: Thanks!

Anon, I think that actually makes sense, but I do have one question. What happens if, in flat (i.e. zero cosmological constant) 2+1 space, you accumulate mass into an object so dense that its escape velocity is greater than the speed of light? Would it simply not be for some reason possible to construct such an object in flat 2+1 space? Or would it be possible for such an object to exist, but the object would not be or behave like a black hole (i.e. no singularity forms)? From what you’re saying it sounds like the latter would be the case.

Well, you’re thinking very intuitively about the situation, which is fine.

However, there’s an important distinction between vacuum and matter solutions and an important relationship as well. In 3+1 there’s a good understanding that collapse of a matter solution will produce a singularity and subsequently an asymptotic final state that is a pure vacuum solution (all matter confined asymptotically to the singularity and so no matter elsewhere; e.g., a pure vacuum solution). The Oppenheimer Snyder analysis points the way / motivates the passage from a matter solution towards a vacuum solution final state, which logically (but not historically) is a primary impetus for looking closely at the 3+1 vacuum space of solutions. Hence consideration of Ricci = 0 solns (vacuum by defn of the Einstein equation). Einstein’s original motivation for looking at Ricci = 0 was different, perhaps explaining why it took the prescient work of Schwarzschild and then Oppenheimer and Snyder after that.

Thus, the motivation for immediately jumping to Ricci = 0 solutions assumes the given of tottal gravitational collapse (which in turn presumes ideas like escape velocity).

Interrogating 2+1 for Ricci = 0 finds a suprise: All vacuum Ricci = 0 are flat.

Hence no total gravitational collapse; hence no analogous escape velocity.

2+1 is different and has special arguments.

HIH

Coin,

The idea of a black hole as an object near which the escape velocity is greater than one is a purely newtonian idea. Laplace famously wrote a paper about it, and someone else before him, but that’s another story. The precise definition of a black hole in general relativity is considerably more complicated than that.

But anyway, let’s think about black holes in 2+1d newtonian gravity. Whereas in 3 spatial dimensions the gravitational force from a point particle goes as the inverse of the square of the distance, in 2 spatial dimensions it just goes as just the inverse of the distance. This is just like the electric field of a line charge.

Anyway, if the force goes as one over the distance, then the binding energy goes as the log of the distance. So it would take an *infinite* amount of energy for a test particle to escape to infinity in 2+1d. Escape velocity isn’t a meaningful concept in 2+1d newtonian gravity, because there just isn’t any escape.

anon writes:

Gotta be a bit careful here. Everything you say is true, but you can have metrics that are vacuum solutions of (2+1)-dimensional general relativity everywhere

exceptfor a singularity. For example, there are solutions where space looks like a cone. A cone is flat except at the tip, where the curvature is infinite. These solutions act like point particles – not black holes.The cool part is that there’s an

upper bound on the mass of the particlein these solutions. More precisely, and even more cool, the mass of a particle is only defined modulo the Planck mass! It makes no sense to talk about a particle with mass greater than the Planck mass… past this point, the mass just ‘wraps around’: a particle like this is the same as a very light particle.By the way, this stuff is true classically: no quantum gravity required! The reason is that in 2+1 dimensions one can define the “Planck mass” – a quantity with units of mass – using just Newton’s constant G and the speed of light c. One doesn’t need Planck’s constant!

All this is much more sensible than it sounds at first sight. For more details, try week232. For even more details, try my paper with Derek Wise and Alissa Crans.

But anyway, yeah: there are no black holes in (2+1)-dimensional gravity when the cosmological constant vanishes… or is positive. One doesn’t need to understand quantum gravity to see this: it’s just GR.

It’s nice to see equations on your blog, Peter! Good, sound, serious stuff – not just people fighting!

Very cool!

Thanks John

coin writes:

This is interesting to do, but general relativity in 2+1 dimensions is funny.

Unlike the more familiar (3+1)-dimensional case, it

does not reduce to Newtonian gravityin the limit of small masses moving much slower than the speed of light!For example, it’s pretty easy to see that in 2+1 dimensions, general relativity does not allow one small mass to orbit another in a circular orbit. Newtonian gravity does.

So, in 2+1 dimensions, ‘Newtonian black holes’ are a poor guide to the behavior of general-relativistic black holes — much worse than in 3+1 dimensions, where John Michell successfully computed what’s now called the ‘Schwarzschild radius’ way back in 1784, using Newtonian gravity! (His calculation was repeated by Laplace around 1795.)

All this makes (2+1)-dimensional general relativity a rather odd subject to use as a warmup for real physics… but it’s very pretty. Most researchers in quantum gravity have not been able to resist its charms — there are lots of papers about it.

“All this makes (2+1)-dimensional general relativity a rather odd subject to use as a warmup for real physics…”

Then why is Witten’s work interesting? I’m not being sarcastic — I genuinely want to know what we might learn if people push on with this thing.

A.V.Pushkin (1947-2004) from Russian Federal Nuclear Centre –VNIIEF (Sarov) used in nineties The Monster Group as hardware for calculations of important dimensionless physical constants, and not only them. Very succinct exposition of his approach you could find in his presentation on the Second Int. Sakharov Conf., Moscow, 1996 (Proc. of this Conf. were published by World Scientific Publ.,1997).Here I place the annotation of this work.

“MONSTROUS MOONSHINE” AND PHYSICS

A.V.Pushkin

The paper presents some results obtained by the author on the quantum gravity theory. This theory proves related to geometry of Cayley projective plane and the algebraic structure of the theory to the commutative nonassociative Griess algebra. The theory symmetry group is the automorphism group of Griess algebra: “Monster” simple finite group. Knowledge of the theory symmetry allows observed physical quantities to be computed in the “zeroth” approximation. Results of the calculations, including those for fine structure constant ~ 1/137 and proton to neutron mass ratio, are presented, with the theory-controlled accuracy of the “zeroth” approximation being higher for some of them by 1–1.5 orders of magnitude than the accuracy of modern measurements.

If it is interesting to somebody I can send full text and additional information.

Gennady Kochemasov, RFNC VNIIEF.

My home e-mail: ggk44@mail.ru..

amanda,

In terms of physics, 2+1 gravity isn’t what one ultimately wants to know about (although, for mathematics, Witten’s claims may make it turn out to be very interesting). It’s a toy model. If one needs to develop new ideas and new techniques for solving a problem (which is what I think quantum gravity needs), then most of the time you have to first start with simpler cases, with toy models, in order to develop these new ideas and techniques. Witten seems to have found a new sort of relationship between a fully quantum gravity theory, gauge theory, and a simple conformal field theory. Yes, it’s in 2+1d, not 3+1d, but it’s a new idea, and until one completely understands how it works and what one can do with it, whether or not it will help with the 3+1d problem is unclear. I think Witten would also argue that sorting out the conceptual problems of a fully quantum, exactly solvable, theory of 2+1d black holes may give insight into the more difficult conceptual problems in 3+1d, where there seems to be no hope of an exact solution.

About the work of Pushkin: this seems to have nothing to do with the topic of this posting, Witten’s recent work, but I’ll leave the comment, suggesting that people who want to discuss that contact the author.

To Peter Woit. I think that your comment about the work of Pushkin and the topic of this posting is not exactly correct, The reason very simple: Pushkin really thought that he solved the problem of quantum gravity and had strong support for such conclusion.If this true(I am not expert in QG, but I had many talks with Sasha), then this posting had projection on wright direction.And it is good news. For explaining my point of view I place here one more citation from Pushkin book(in Russian):“In the superstring theory a striking progress has been achieved recently: as few as five theories remain which are actually based on two lattice types in 16-dimensional space. Moreover, it has been proved that they are non-perturbative equivalents and as such are the limiting cases of the more general single theory. What last step should be made in the superstring theory in the direction of the construction of a single theory? Of course, this step is the gravity quantization, which will result in disappearance of the last dimensional scale in the theory, viz. the fundamental string scale. Then the theory will possess the local scale invariance properties and may perform quantitative calculations with non-perturbative methods by translating algebraic relation between physical quantities to different scale levels, including to the level of ordinary phenomena appearing in continuum motions in standard conditions.” Here I pass long but important part of text and proceed:”In geometrodynamics, i.e. upon the gravitation quantization, even this last dimensional scale disappears. In the language of the lattice theory in spaces this means that there should be the only possibility to avoid the need of appearance of dimensional scale: the fundamental root amount is equal to infinity (i.e. there is no separated cell) and the Weyl vector is therewith light-like, which also requires no a priori scale. It is evident that the last requirement is readily formalizable at the level of the elementary problem of Diophantine equation solutions . Having solved it, you will find those values of dimension of hyperbolic spaces , which determine the lattice arrangement of the geometrodynamics.”

It may be helpful to emphasize, as Peter mentioned above, that

a) there is a well-known relation between 2-dimensional conformal field theory and 3-dimensional Chern-Simons theory for a given gauge group G. This is a deep and powerful relation between two classes of field theories.

b) it so happens that for particular choices of gauge group, Chern-Simons theory becomes equivalent to 2+1 dimensional gravity.

This is probably a “coincidence of low dimensions”: two classes of structures which are by themselves rather different happen to coincide when both are restricted to sufficiently degenerate cases (here: gravity restricted to sufficiently low dimensions), simply because there is not enough room for them to differ:

here both the Chern-Simons action and the Einstein-Hilbert action (in its Palatini formulation) must be suitably invariant functionals pairing a connection 1-form and a 3-manifold. There are not that many possibilities for that, in a sense.

So the idea is: (quantum) gravity in general is an awefully complex issue, which we arer having a hell of a hard time coming to grips with. After we are sufficiently frustrated about our lack of progress in the general case (or even the one case, 3+1 dimensions, which we started off being interested in), we fall back to the observation that if only we constrain one parameter (the number of dimensions) sufficiently, suddenly everything becomes tractable.

Before we embarked on the quest for quantum gravity we had been optimistic that progress is possible for the cases that we were really a priori intersted in and didn’t care about that unphysical 2+1-dimensional case.

But now that frustration with the general case is high enough, the case we previously looked down on suddenly appears much more charming.

So we say: “Let’s then at least fully work out everything in that toy example. Better than getting nowhere.”

And then it turns out that even the toy example is demanding enough and that we should maybe have trained our skills at this from the very beginning.

Hopefully, someone will correct me.

In 3+1 there are some deep problems with quantum gravity. One of which is the problem of time and others related to the problem of quantifying the geometry (or the arena) on which physics occurs.

Some writers originally approached 2+1 as a ‘laboratory’ (or as Peter calls it a toy model) where problems like the problem of time or the quantization of geometry could be considered towards the end of generating intuition / ideas about the 3+1 problem.

it so happens that for particular choices of gauge group, Chern-Simons theory becomes equivalent to 2+1 dimensional gravity.As I understand it, this is not at all clear. It is true that there are classical transformations that take one to the other, but the relation of the quantum theories (whatever they may be) is not obvious.

As I understand it, this is not at all clear. It is true that there are classical transformations that take one to the other, but the relation of the quantum theories (whatever they may be) is not obvious.

Ah, is it not that Witten in his approach effectively

defines2+1D quantum GR to be G-Chern-Simons, for suitable G?(Hm, in the preview of the above comment the nested blockquote did work. )

Indeed, even the classical theories are not equivalent, since in the CS formulation degenerate metrics are included. As a consequence, the gauge equivalence classes of Einstein-Hilbert gravity and “CS gravity” are not the same, see for instance gr-qc/9903040 for an elaborate discussion.

As Urs points out, Witten, in his -88 paper, rather takes the CS formulation as the definition of 2+1D quantum gravity

One has to just take some definition of 2+1-dimensional quantum gravity, study it, and see if ‘acts like a quantization of general relativity in 2+1 dimensions should’ — bearing in mind that general relativity in 2+1 dimensions has a lot of weird features, and quantization will make things even weirder!

There are a number of BF theories and Chern–Simons theories that have a claim to describing gravity 2+1 dimensions, depending on the signature of the metric and the value of the cosmological constant. Derek Wise will give a thorough listing of these in his thesis (due in a few weeks).

As Jens notes, all these theories allow degenerate metrics, unlike Einstein-Hilbert gravity. That may be okay; one just need to see what happens, especially upon quantization.

The more realistic theories have Lorentzian signature, so they have noncompact gauge groups, which introduce a lot of technicalities that people don’t fully understand yet. For example: the representation theory of noncompact quantum groups and noncompact affine Lie algebras.

For those you want to take a look on a simple example (among many) of 2+1 Gravity and 2-d CFT relation, you can take a look here [hep-th/0411060].

This papers starts with 2+1 CS gravity as used by Witten and derives 2-d Ward identities which are characteristic from CFT and known for long time. The key is to considers proper boundary terms, inpired by AdS/CFT correspondence.

One way to think of a realization of the relation

G-WZW theory on 2D-bundary G-CS theory on 3D-bulk

seems to be to think of membranes in a Horava-Witten setup:

the membrane, propagating in some spacetime, couples to the supergravity 3-form C, which looks like

C = CS(A_e8) + CS(A_so(10,1)) + dB,

where A_e8 is an e_8 connection and A_so(10,1) an so(10,1)-connection and CS(…) are the corresponding Chern-Simons terms.

Varying the membrane configurations is a lot like varying the pulled back connections A_…, hence a lot like variation of a Chern-Simons functional on the membrane.

So we might expect a corresponding WZW theory on the boundary of the membrane. And indeed, for the e8 part we do: that should be the internal current algebra CFT of the heterotic string.

But what about the SO(10,1)-part?

At least naively, it should somehow split, locally, into the SO(2,1)-part tangential to the membrane and an SO(8)-factor transversal to it.

Possibly the SO(8)-WZW model may be merged with the E_8 one some way or other. What would be left is an SO(2,1)-CS theory on the membrane. Since that’s, roughly, workvolume gravity — as discussed in the above thread — , it looks a little like part of the worldvolume kinetics of the membrane.

I am just saying this in the hope that somebody out there recognizes something in these observations that can actually be made a little more precise. I have here three Lie 3-algebras corresponding to the above three ingredients lying around, and I would like to fuse them back to the one thing they came from. It vaguely looks like understanding the above should help understand this problem. Or vice versa.

Urs,

Do you really believe that a good way to think about the relation between the simplest 2d conformal field theories (WZW models), and simple 3d QFT with a single, beautiful term in the action (CS) is to embed everything in 11 dimensions, and try and relate things to 11d supergravity and an unknown M-theory?

Yes.

By the way, maybe in what I wrote above I am being stupid with seeing a problem here where the formalism is trying to tell me that there is no problem:

On p. 79 of Wiaeo? the authors define a Chern-Simons theory parameterized by a G-bundle E->X to involve only those bundles with connection, which can be obtained from pullback of E along some map Sigma : M–>X. Then one only varies those maps Sigma.

That’s

preciselythe situation we have for the 3-form coupling of the membrane!Hm…

It is really nice to read unpublished newest ideas in physics as blog and to have a chance to post a comment though I am a simply beginner of string theory.

I cannot tell well but I am wondering newest 2+1 D doesnot mean usual spatial 2D + time D but spatial D + time D + extra D (or spin geometry?).

Regards,

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I say a bit more about Witten’s new idea in week254.

In particular, since Urs and some of his collaborators know a lot about how to get 3d topological quantum field theories from 2d rational conformal field theories, some of them may enjoy trying to extract a 3d TQFT from the c = 24 conformal field theory Witten is considering as dual to 3d quantum gravity in his new work – the one that has the Monster group as symmetries.

If Witten’s ideas are correct, this 3d TQFT could be a kind of “improved version” of Chern-Simons theory when it comes to describing 3d quantum gravity. This seems like a strange idea, but perhaps interesting.

(I’m not sure this field theory is a rational conformal field theory, or has been proved to be so. I believe Borcherds, Frenkel, Lepowsky and Meurman only consider it as a vertex operator algebra. I hope it extends to a rational CFT. But, this could be hard to show, even if it’s true.)