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Pariah moonshine

Erica Klarreich wrote an insightful review

Moonshine Link Discovered for Pariah Symmetries (Quanta Mag.)
of a new paper by Duncan, Mertens, and Ono in Nature,
Pariah moonshine (full paper, HTML).
That discovery is a counterpart of the monstrous and umbral moonshine – but instead of the monster group and umbral/mock modular forms, it deals with a pariah group and weight 3/2 modular forms.

The historical bottles of Old Hunter's, a Czech whiskey, indicate that the hunter was getting younger as a function of time. ;-)

The paper was originally sent to me by Willie Soon – who wasn't the only one who was entertained by the terminology. This portion of mathematics really uses very weird or comical jargon, maybe one that is over the edge. But I believe that the playful names ultimately reflect the unusual degree of excitement among the mathematicians and mathematical physicists who study these things – and I believe that this excitement is absolutely justified.

I don't want to cover their discoveries in detail but it may be a good idea to remind you of the three kinds of moonshine and how big a portion of ideas they cover.

OK, in all cases, there's something like a big group or a higher-dimensional geometry with some topology on one side; and some possibly generalized modular form on the other side. A modular form is basically a holomorphic function of usually one complex variable \(\tau\) that has some simple enough behavior if you change \(\tau\to \tau+1\) or \(\tau\to -1/\tau\), if you allow me this non-rigorous but arguably very helpful heuristic definition.

Some large, integer coefficients seem to be the same on both sides. Because the two sides look like coming from completely different corners of mathematics, you're tempted to think that the match must be a coincidence. Except that, as you may realize already shortly before you drink a bottle of moonshine that you win for your proof, the agreement isn't a coincidence at all. There exists a mathematical explanation why the numbers in the different corners of mathematics have to be equal.

This explanation is generally referred to as moonshine.

The first and still deepest – I hope that experts agree – example of moonshine is the monstrous moonshine which includes the monster group as the main actor on the geometric side. (Check for my introduction to moonshine at Quora if you're intrigued.) The monster group is the largest among the truly exceptional, so-called sporadic, finite groups. It has almost \(10^{54}\) elements. The order is the product of powers of all supersingular primes – there is a finite number of supersingular primes (all primes up to 71 except for 37,43,53,61,67) which are "more prime" than other primes.

The monster group has irreducible representations. The smallest one is obviously the 1-dimensional trivial one that doesn't transform at all. The next one is 196,883-dimensional. It turns out that the \(j\)-invariant, a unique (up to \(SL(2,\CC)\) transformations) holomorphic function mapping the fundamental region of the modular group to the whole complex plane (therefore a weight-zero modular form), may be expanded as\[

j(\tau) = {1 \over q} + 744 + 196884 q + 21493760 q^2 + \dots

\] for \(\tau\to i\infty\) where \(q=\exp(2\pi i \tau)\). The coefficient 196,884 looks similar to the dimension of the irrep of the monster group and indeed, it's no coincidence. There exists a string theory – basically the dynamics of ordinary string theory's strings propagating on a 24-dimensional torus defined as a quotient of \(\RR^{24}\) by the Leech lattice, the nicest among 24 different even self-dual 24-dimensional lattices (it's nicest because it happens to have no sites whose squared length is two, the positive minimum allowed by the even self-dual condition).

One can show that the partition sum (on the torus) of this string theory has to be basically the \(j\)-function. How? Partition sums in string theory basically have to be some modular functions and due to the absence of the sites I mentioned and some other simple arguments, the function must be weight-zero and those are basically unique. At the same moment, the string theory may have be proven to have a discrete symmetry which is a stringy extension of the "obvious" isometry group of the Leech lattice. The automorphism group of the Leech lattice is the Conway \({\rm CO}_0\) group and string theory enhances it to the full monster group \(M\).

So all the coefficients in the \(j\)-functions have to count some degeneracy in a string theory but because the string theory has a monster group symmetry, all these degeneracies have to come in full representations of the monster group. At low energy levels, the smallest representations appear, and \(196,884\) in the expansion of the \(j\)-invariant has to be \(1+196,883\), the dimension of the direct sum of the two smallest irreps. The numbers at higher levels are some simple combinations of lower-dimensional irreps of the monster group, too.

Great, so string theory on the Leech torus – in some sense, the coolest way to compactify all 24 transverse dimensions of the bosonic string – explains the monstrous moonshine.

Then there is another moonshine, umbral moonshine. It is a generalization that replaces the monster group by the Mathieu group \(M_{24}\) or by something that is slightly more general. The relevant string theory includes strings propagating on the K3 surfaces. Because the "total" homology of the K3 surfaces is 24-dimensional, this moonshine is also related to 24-dimensional lattices – but all 24 even self-dual ones (Niemeier lattices), not just the Leech lattice.

The generalization of this Mathieu moonshine is called "umbral moonshine" because "umbral" is a Latin adjective derived from "shadows" and the relevant modular functions aren't really true modular functions but mock modular functions, some generalizations – and generalizations like these mock ones may be called "shadows". See TRF blog posts with "umbral".

The integers that appear in the umbral moonshine are typically smaller but there are many of them and they may be matched in between the two sides in some way.

The newest type of moonshine which we discuss here is the pariah moonshine. So far the string theory compactification isn't really known, if I understand well, and there should be one so the readers are urged to immediately find it! What is "pariah" about it?

Well, look at this list of sporadic groups. They were named after the generalized mathematicians who discovered them – a generalized mathematician is defined as either a smart man, a monster, or a baby monster. ;-) The mathematicians after whom these exceptional finite simple groups which don't fall into any infinite families – the sporadic groups – are Mathieu (5 of them), Janko (4), Conway (3), Fischer (3), Higman-Sims, McLaughlin, Held, Rudvalis, Suzuki, O'Nan, Harada-Norton, Lyons, Thompson, baby monster, and monster (equivalently Fischer-Griess). That's 26 sporadic groups in total.

The Tits group is sometimes counted as the 27th sporadic group – it's "almost" a group of the Lie type, because of some subtleties.

OK, the diagram above shows the 26 full-blown sporadic groups. The relationships indicate how you can get the groups from each other as subquotients. Most of them end up in a herd, underlied by the monster \(M\) at the top, and those are called "the happy family" which has 20 members. The remaining 6 sporadic groups are the four groups on the right bottom side plus the Lyons and Janko-4 group on the top sides. These 6 groups were named the pariah groups – the opposite of the happy family had to be invented. All the 26 sporadic groups are exceptions or renegades of a sort – the pariah groups are arguably even more blasphemous, they're the renegades among heretics. ;-)

The new paper in Nature is mostly about the O'Nan sporadic group – although some related constructions probably exist for \(M_{11}\) and \(M_{23}\) near the bottom, descendants of the monster in the happy family. The authors chose the adjective "pariah" for the moonshine but they didn't immediately say something about all the pariah groups, I think. So maybe a better title could have been "O'Nan moonshine" – which sounds like some truly original Irish whiskey. ;-) (Recall that the Irish whiskey is spelled with "e" while the Scottish whisky is spelled without "e". Michael O'Nan was American and died at Princeton on July 31nd. Later, I learned that the authors did use the O'Nan moonshine phrase on the arXiv.)

The counterpart of the monster group's irrep's dimension 196,883 happens to be 26,752 (see the full list) for the O'Nan group and you may see it as a coefficient in weight-3/2 modular functions. That's a basic connection that this pariah moonshine is all about.

In some very vague sense, these moonshines are analogous to each other – recently, weight-1/2 modular functions were connected with the Thompson group, a member of the monster's happy family – but there seem to be rather deep technical, almost groundbreaking differences between the individual members of the community. What I want to say is that it would be totally wrong to impose some "egalitarianism" between the 26 sporadic groups and think that the knowledge about their moonshines is composed of 26 copies of the same clichés. Each of them is very different, has connections to somewhat different parts of mathematics, group theory, and string theory. Each of them has a slightly different story. Affirmative action would be totally indefensible here.

Above, I wrote that there were basically 3-4 moonshines – if you attach the major adjectives such as monsterous, Mathieu, umbral, and pariah. But with a finer technical classification, the number is larger. By 2013, Miranda Cheng, John Duncan, and Jeff Harvey conjectured the existence of 23 new moonshines – and Michael Griffin proved their existence two years later.

These moonshines are a deep wisdom in mathematics – mental wormholes that connect completely different regions of the realm of mathematical ideas. They may also be interpreted as dualities in string theory. In particular, the monstrously symmetric compactification of string theory defines the CFT that is holographically dual to pure gravity in the three-dimensional anti de Sitter space. Similarly, the other sporadic groups – the most complicated and exceptional groups in the theory of finite groups – are being connected to some of the simplest and most fundamental geometric compactifications of string theory. I think that there clearly seems to be a new complementarity here – a fundamental, structureless compactification of string theory is linked to the largest sporadic groups. You make the stringy compactification a little bit more arbitrary and the corresponding moonshine's group gets smaller or more regular.

Very generally, I have often said and some other people have said that string theory isn't just a theory of everything in physics. It may also provide us with a complete classification of all ideas in mathematics that are really worth something. I think that the moonshines and perhaps some related ideas are very particular and important manifestations of this power of string theory to classify and connect all profound mathematical ideas.

Again, I write it despite the fact that the stringy compactification explaining the pariah moonshine hasn't been found yet – in this sense, the proof of the pariah moonshine seems to be non-stringy, at least so far. The stringy compactification for this case could be a minor variation of the known ones – and maybe of M-theory or F-theory which would already be cool – but it could be more fundamental and more original, too. We should better find it or whatever replaces it. (Don't expect it to be a loop quantum gravity or another "very non-stringy" or crackpot-powered beasts, however LOL.)

By the way, at this moment, the Quanta Magazine review has only two comments – and both of them say that the readers have no chance to understand it. I am afraid that it's really something like that proverbial "twelve people in the world" who can really work on these things technically and even the number of people who understand it at the level of your humble correspondent is of order 100 in the world. I have mixed feelings about the question whether the number should be substantially higher, and whether it makes any sense to write popular reviews (and give popular talks about moonshine, as I did in the summer) at all.

There are some general prejudices that the laymen have about the "details of mathematical structures". Some people are numerologists – they are obsessed with seeing special numbers like 137 everywhere and invent lots of new moonshines which are rubbish, of course. They're enthusiastic cranks – their fantasy runs too hot. But most laymen are the exact opposite. They are too cold and they believe that there's nothing special to be seen anywhere. They basically think that all numbers are created morally equal. If something can be done with 248, it can be done with 252, too. Well, it isn't the case. The truth of mathematics is somewhere in between. Lots of very particular statements and structures exist that are tightly connected to some special integers and simply cannot be generalized to all integers. All the numbers describing the sporadic groups – and moonshines – are great examples of these exclusive rights and inequality between the numbers and choices in mathematics. I think that this is the kind of a very general, moral conclusion that the mathematical research has achieved and that hasn't been communicated to the public yet – a reason to think that the popularization isn't really working at all.

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