By pointing out the absence of some twist fields using group theory arguments, Gaiotto has shown that counterparts of this theory don't exist for every \(N\) – a problem arises already at \(N=2\). But even if there only exists the \(N=1\) theory with the minimal \(AdS_3\) curvature radius, it's still interesting. Perhaps exceptionally interesting. It may be the most "negatively curved" \(AdS\) background in all of quantum gravity, among other things.

The holographic analyses often focus on the \(AdS_3\) part of the bulk geometry. So the assumption is that only the \(AdS_3\) geometry is "real" in the bulk. But I think that this is a superficial and uninspired treatment – and a similar oversimplification as if you said that the dual geometry to the \(\NNN=4\) gauge theory is the \(AdS_5\). Well, there's also the \(S^5\) in \(AdS_5\times S^5\). In fact, both factors have exactly the same curvature radius so if you don't neglect one, you shouldn't neglect the other, either.

The monster group includes a huge amount of internal structure. It's the largest among the 26 or 27 sporadic groups in the theory of finite groups. It has \(8\times 10^{53}\) elements, 194 conjugacy classes (and therefore 194 irreducible representations), and at least 44 conjugacy classes of maximal subgroups.

The degeneracies of the black hole microstates in the \(AdS\)/CFT pair increase sort of exponentially, as you expect from black hole thermodynamics. Papers usually don't try to "imagine" any structure at a given level. The levels may at most be divided to representations of the monster group.

But in the \(AdS_5\times S^5\) case, there's a lot of structure. Objects may be moving along the \(S^5\), strings may vibrate there, strings and branes may wrap submanifolds of the \(S^5\), and so on. There's some emergent locality not only in the \(AdS_5\) directions but in the \(S^5\) directions of the type II stringy spacetime, too.

Although I still don't understand the full mathematics, I think that there's a similar story of "truly quantum geometry" behind the monster group. Let us look at some analogies between the \(AdS_3\) and \(AdS_5\times S^5\) vacua.

**Global symmetry, hidden manifolds**

First, the \(\NNN=4\) gauge theory has a global symmetry, \(SO(6)\). This is the R-symmetry rotating the supercharges into each other. In the bulk, this group is reinterpreted as the bulk gauge group which rotates the 5-sphere as an isometry – the gauge group arises from the non-Abelian Kaluza-Klein mechanism. In fact, the group \(SO(6)\) almost directly determines the geometry of the compact extra dimensions:\[

SO(6) / SO(5) \approx S^5

\] So if you take the quotient of \(SO(6)\) by a maximal subgroup, \(SO(5)\) in this case, you just get the compact manifold that is hiding in the bulk, the five-sphere. Shouldn't the same thing work for the monster group? I say it could be the same thing but it's not "quite" the same thing, e.g. because the global symmetry of the monstrous CFT is a finite group, not a Lie group.

Just to remind you, \(SO(5)\) isn't the only maximal subgroup of \(SO(6)\). Another maximal group is \(U(3)\) which is embedded by interpreting complex numbers as real \(2\times 2\) matrices. The corresponding quotient\[

SO(6) / U(3) \approx {\mathbb C \mathbb P}^3

\] would be the projective space. The natural metric on the projective space that you get by constructing it as this coset is the so-called Fubini-Study metric. Stringy compactifications on the projective space exist – in fact, they're also very important in the membrane minirevolution a decade ago.

Now, take the monstrous global symmetry. It's finite but it's the global symmetry of the boundary CFT. By analogy, it should correspond to the gauge group of the bulk theory. If the analogy really works well, there could be a hidden "manifold" – probably represented by a collection of points\[

M/G

\] where \(G\) is a maximal subgroup of \(M\), the monster group. Maybe several maximal groups could be used at the same moment, who knows. Now, there are at least 44 ways to choose the maximal subgroup \(G\) in unequivalent ways. The maximal subgroups involve many other types of other sporadic groups, infinite families of finite groups based on the usual Lie groups (exceptional and unexceptional ones) above various fields, permutation groups, and others.

There should be some sense in which \(M/G\) is the "compact manifold" of the bulk spacetime whose other factor is the \(AdS_3\). Such a quotient should be an approximation of a smooth manifold in some sense of the "approximation" – assuming some proper interpretation of the "approximation". Needless to say, such a manifold would behave as an example of the quantum geometry.

**Volume of the manifold**

If geometry is composed of individual points, the "volume" is dimensionless and it's numerically equal to the number of points. The quotients \(M/G\) have the numbers of elements that may be written as products of powers of the supersingular primes (primes up to 71 excluding 37, 43, 53, 61, 67: those appear in the factorization of the order of the monster group and may be defined in many, surprisingly different ways, too).

These often large numbers may be "approximations" of some numbers of order one. How could it work? Well, products of very many primes may behave like that. For example, if you have the product of all primes, you may write\[

2\times 3 \times 5 \times 7 \times \dots = 4\pi^2

\] It's "four pi squared" in a similar sense in which\[

1+2+3+4+5+\dots = - \frac{1}{12}.

\] In fact, some of the same regularization techniques may be exploited to calculate both results. Fine. The product of all primes is a rational multiply of a power of \(\pi\). Note that the product of all positive integers is\[

1\times 2 \times 3 \times 4 \times \dots = \infty! = \sqrt{2\pi}.

\] The regularized value is the same \(\sqrt{2\pi}\) that you may see in Stirling's approximation for the factorial. All the other "simple and divergent" factors are regularized to one.

If \(M/G\) had the volume (number of points) that would be the "product of almost all primes", we could say that the volume is "almost \(4\pi^2\)". What would it tell us? It would tell us that the compact manifold may be something analogous to a sphere. Note that the 3-volume of a unit \(S^3\) is \(2\pi^2\), one-half of the product of all primes.

There could be an approximate notion of locality on \(M/G\). The approximation could be good because "we have added primes except for some negligible – very large – ones". The omnipresence of products over primes could mean that the amplitudes emerge as products from some \(p\)-adic factors. Recall that the Veneziano amplitude may be written as the inverse of the product of all Veneziano amplitudes in corresponding \(p\)-adic string theories where \(p\) are primes.

Can you find mathematicians' papers that combine \(p\)-adic and sporadic objects? You bet, see e.g. Ono, Rolen, Trebat-Leder. By the way, is it a coincidence that sporadic and \(p\)-adic end with the same "adic"? ;-) I can't follow the paper by Ono et al. too much because it's too mathematical or unphysical for me. But papers like theirs could store some physical wisdom that hasn't been uncovered yet.

Recall that I also believe that quantum gravity links the total volume or number of elements of the symmetry group to some simple expressions and there's a complementarity between the "size of the symmetry group" and the "complexity of the spacetime geometry". The monster group \(M\) of the \(AdS_3\) vacuum is so huge exactly because the geometry of \(AdS_3\) is so simple or irreducible.

These things are ambitious and a big discovery about "how quasi-smooth geometry may really arise from points and finite symmetries and primes etc." isn't guaranteed. But I think that at least dozens if not hundreds of people on this big blue planet – hopefully many people who know the maths much better than I do – should be thinking about those matters with me, along with other strategies to crack the deepest secrets about quantum geometry and the theory of everything.

It seems plausible that lots of things that have been done with \(AdS_5\times S^5\) may be done with \(AdS_3\times X\), too. For example, there could be a pretty good Penrose \(pp\)-wave limit, analogous to the BMN limit of the gauge theory. Highly excited black holes in the monstrous background could be constructible as some counterparts of strings, perhaps some spin networks based on the monster group symmetry (or the Chern-Simons theory based on the monster group).

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