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Pure \(AdS_3\) gravity from monster group spin networks

A fifth of my research topics that make me most excited have something to do with the three-dimensional pure Anti de Sitter space gravity. In 2007, Witten pointed out that there is a perfect candidate for the dual boundary CFT, one that has the monster group as the global symmetry.

The monster group is the largest among the 26 or 27 "sporadic groups" in the classification of all the simple finite groups. The CFT – which was the player that proved the "monstrous moonshine" – may be constructed from bosonic strings propagating on the (24-dimensional space divided by) the Leech lattice, the most interesting even self-dual lattice in 24 dimensions, the only one among 24 of those that doesn't have any "length squared equals to two" lattice sites.

I didn't have enough space here for a picture of Witten and a picture of a monster so I merged them. Thanks to Warren Siegel who took the photograph of Cyborg-Witten.

The absence of these sites represents to the absence of any massless fields. So the corresponding gravity only has massive objects, the black hole microstates, and they transform as representations of the monster group. I will only discuss the monster group CFT with the "minimum radius" – Davide Gaiotto has proven that the infinite family of the larger CFTs cannot exist, at least not for all the radii and with the required monster group symmetry, because there's no good candidate for a spin field corresponding to a conjugacy class.

I think that the single CFT with the single radius is sufficiently fascinating a playground to test lots of ideas in quantum gravity – and especially the relationship between the continuous and discrete structures (including global and gauge groups) in the bulk and on the boundary.

It's useful to look at the list of irreducible representations of the monster group for at least 10 minutes. There are 194 different irreps – which, by Schur's tricks, means that there are 194 conjugacy classes in the monster group. Don't forget that the order of any element has to be a supersingular prime.

However, you will only find 170 different dimensionalities of the irreps. For years ;-), I have assumed that it means that 146 dimensionalities are unique while for 24 others, the degeneracy is two – so the total number of irreps is 146*1+24*2 = 194. It makes sense to think that some of the representations are complex and they're complex conjugate to each other, in pairs.

Well, just very very recently ;-), I looked very very carefully, made a histogram and saw that one dimension of the irreps, namely the dimension

5 514 132 424 881 463 208 443 904,
(5.5 American septillions) appears thrice – like the 8-dimensional representation of \(SO(8)\) appears in "three flavors" due to triality. Why hasn't anyone told me about the "tripled" irrep of the monster group? I am sure that all monster minds know about this triplet of representations in the kindergarten but I didn't. So the right answer is that there are 147 representations uniquely given by their dimension, 22 dimensionalities appear twice, and 1 dimensionality (above) appears thrice.

BTW those 5.5 septillions has the factor of \(2^{43}\) – a majority of the factor \(2^{46}\) in the number of elements in the monster group – and no factors of three. This large power of two is similar to the spinor representations (e.g. in the triality example above).

Fine. Among the 194 irreps, there's obviously the 1-dimensional "singlet" representation. Each group has singlets. The first nontrivial representation is 196,883-dimensional. This many states, along with a singlet (so 196,884 in total), appear on the first excited level of the CFT – so there are 196,884 black hole microstates in pure \(AdS_3\) gravity with the minimum positive mass (this number appears as a coefficient in an expension of the \(j\)-invariant, a fact that was known as the "monstrous moonshine" decades before this "not so coincidence" was explained). This level of black holes has some energy and nicely enough,\[

196,883\sim \exp(4\pi),

\] as Witten was very aware, and this approximate relationship is no coincidence. So the entropy at this level is roughly \(S\approx 4\pi\) which corresponds to \(S=A/4G\) i.e. \(A\approx 16\pi G\). Note that the "areas" are really lengths in 2+1 dimensions and Newton's constant has the units of length, too. The entropy proportional to \(\pi\) is almost a matter of common sense for those who have ever calculated entropy of 3D black holes using stringy methods. But it's also fascinating for me because of my research on quasinormal modes and loop quantum gravity.

The real part of the asymptotic, highly-damped quasinormal modes was approaching\[

\frac{\ln 3}{8\pi G M}

\] where \(M\) is the Schwarzschild black hole mass. The argument \(3\) in the logarithm could have been interpreted as the degeneracy of some links in \(SO(3)\) spin networks – and that's why I or we were treated as prophets among the loop quantum gravity and other discrete cultists and why my and our paper got overcited (although we still loved them). It's a totally unnatural number that appears there by coincidence, and I – and I and Andy Neitzke – gave full analytic proofs that the number is \(3\) exactly. It's not a big deal, it's a coincidence, and \(3\) is a simple enough number so that it can appear by chance.

But the funny thing is that the quasinormal frequency becomes a more natural expression if \(3\) is replaced with another dimension of an irrep. Fifteen years ago, I would play with its being replaced by \(248\) of \(E_8\) which could have been relevant in 11-dimensional M-theory, and so on. (\(E_8\) appears on boundaries of M-theory, as shown by Hořava and Witten, but is also useful to classify fluxes in the bulk of M-theory spacetimes in a K-theory-like way, as argued by Diaconescu, Moore, and Witten. Note that "Witten" is in all these author lists so there could be some extra unknown dualities.) And while no convincing theory has come out of it, I still find it plausible that something like that might be relevant in M-theory. The probability isn't too high for M-theory, however, because M-theory doesn't seem to be "just" about the fluxes, so the bulk \(E_8\) shouldn't be enough to parameterize all of the physical states.

But let's replace \(3\) with \(196,883\) or \(196,884\), the dimension of the smallest nontrivial irrep of the monster group (perhaps plus one). You will get\[

\frac{\ln 196,883}{8\pi G M} \approx \frac{1}{2GM}

\] The \(\pi\) canceled and the expression for the frequency dramatically simplified. Very generally, this nice behavior may heuristically lead you to study Chern-Simons-like or loop-quantum-gravity-like structures where the groups \(SU(2)\) or \(SO(3)\) or \(SL(2,\CC)\) which have 3-dimensional representations is replaced with the discrete, monster group.

A fascinating fact is that aside from this numerological observation, I've had numerous other reasons to consider Chern-Simons-like theories based on the finite, monster group. Which ones?

Well, one reason is simple. The boundary CFT of Witten's has the monster group as its global symmetry. So the monster group is analogous e.g. to \(SO(6)\) in the \({\mathcal N}=4\) supersymmetric gauge theory in \(D=4\) which is dual to the \(AdS_5\) vacuum of type IIB string theory. Just like the \(SO(6)\) becomes a gauge group in the bulk gravitational theory (symmetry groups have to be gauged in quantum gravity theories; this one is a Kaluza-Klein-style local group), the monster group should analogously be viewed as a gauge group in the \(AdS_3\) gravitational bulk.

On top of that, there are gauge groups in \(AdS_3\) gravity. In 1988, the same Witten has showed the relationship between the Chern-Simons theory and 3D gravity. It was a duality at the level of precision of the 1980s although decades later, Witten told us that the duality isn't exact non-perturbatively etc. But that Chern-Simons theory replacing the fields in 3-dimensional gravity could be correct in principle. Just the gauge group could be incorrect.

Well, maybe it's enough to replace \(SL(2,\CC)\) and similar groups with the monster group.

One must understand what we mean by a Chern-Simons theory with a discrete gauge group and how to work with it. Those of us who are loop quantum gravity experts ;-) are extremely familiar with the spin network such as

This is how the LQG cultists imagine the structure of the 3+1-dimensional spacetime at the Planckian level. There is obviously no evidence that this is the right theory, nothing seems to work, nothing nice happens when the 4D gravity is linked to those gauge fields in this way, no problem or puzzle of quantum gravity is solved by these pictures. But the spin networks are still a cute, important way to parameterize some wave functionals that depend on a gauge field. Well, I guess that Roger Penrose, and not Lee Smolin, should get the credit for the aspects of the spin networks that have a chance to be relevant or correct somewhere.

If you consider an \(SU(2)\) gauge field in a 3-dimensional space, you may calculate the "open Wilson lines", the transformations induced by the path-ordered exponential of the integral of the gauge field over some line interval. It takes values in the group itself. As an operator, it transforms as \(R\) according to the transformations at the initial point, \(\bar R\) according to the final point – you need to pick a reference representation where the transformations are considered. And you may create gauge-invariant operators by connecting these open Wilson lines – whose edges are specified by a transformation – using vertices that bring you the Clebsch-Gordan coefficients capable of connecting three (or more) representations at the vertex.

Above, you see a spin network. The edges carry labels like \(j=1/2\), the non-trivial irreps of \(SU(2)\). They're connected at the vertices so that the addition of the angular momentum allows the three values of \(j\) to be "coupled". For \(SU(2)\), the Clebsch-Gordan coefficients are otherwise unique. Each irrep appears at most once in the tensor product of any pair of irreps.

Now, my proposal to derive the right bulk description of the \(AdS_3\) gravity is to identify an \(SO(3)\) Chern-Simons-style description of the 3D gravity and replace all the 3-dimensional representations – in \(SO(3)\), the half-integral spin irreps are prohibited – with the monster group.

In this replacement, it should be true that a majority of the edges of the spin network carry \(j=1\) i.e. the 3-dimensional representation. And that 3-dimensional representation is replaced with the \(196,883\)-dimensional one in the monster group case. Otherwise the structures should be analogous. I tend to believe that the relevant spin networks should be allowed to be attached to the boundary of the Anti de Sitter space, and therefore resemble something that is called Witten's diagrams – the appearance of "Witten" seems like another coincidence here ;-) because I don't know of good arguments (older than mine) relating these different ideas from Witten's assorted papers.

Note that the 196,883-dimensional representation is vastly smaller than the larger irreps: the next smallest one is 21-million-dimensional, more than 100 times larger. And it's also useful to see how the tensor product of two copies of the \(d_2=196,883\)-dimensional irrep decompose to irreps. We have:\[

d_2^2 = 2(d_5+d_4+d_1) + d_2.

\] Both sides are equal to 38,762,915,689, almost 39 billion. So the singlet appears twice, much like the fifth and fourth representation. But the same 196,883-dimensional representation appears exactly once (and the third, 21-million-dimensional one is absent). It means that there's exactly one cubic vertex that couples three 196,883-dimensional representations. On top of that, because of the "two singlets" \(2d_1\) on the right hand side above, there are two ways to define the quadratic form on two 196,883-dimensional representation.

I think that in some limit, the spin networks with the "edges 196,883" only will dominate, and the extra subtlety is that each of these edges may or may not include a "quadratic vertex" that switches us to the "less usual" singlet among the two. The presence or absence of this quadratic vertex could basically have the same effect as if there were two different types of the 196,883-dimensional irrep, unless I miss some important detail which I probably do.

Now, there might exist a spin-network-like description of the black hole microstates in \(AdS_3\) and the reason why it works could be a relatively minor variation of the proof of the approximate equivalence of the Chern-Simons theory and the three-dimensional general relativity. The mass of the black hole microstates could be obtained from some "complexity of the spin network" – some weighted number of vertices in the network etc. which could follow from the \(\int A\wedge F\)-style Hamiltonians.

I believe that according to some benchmarks, the \(AdS_3\) pure gravity vacuum should be the "simplest" or "most special" vacuum of quantum gravity. The gauge group is purely discrete which is probably an exception. That's related to the complete absence of the massless fields or excitations which is also an exception. And things just should be more or less solvable and the solution could be a clever variation of the equivalences that have already been written in the literature.

If some deep new conceptual principles are hacked in the case of the monstrous \(AdS_3\) gravity, the remaining work needed to understand the logic of all quantum gravity vacua could be as easy as a generalization of the finite group's representation theory to Lie groups and infinite-dimensional gauged Lie groups. Those also have irreps and conjugacy classes and the relationships between those could be a clever version of the proof that the old matrix model is equivalent to free fermions. Such a unified principle obeyed by all quantum gravity vacua should apply to spacetimes, world sheets, as well as configuration spaces of effective field theories.

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