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Umbral moonshine and Golay code

Many of us greatly liked Erica Klarreich's article

Mathematicians Chase Moonshine’s Shadow
in the Quanta Magazine. The subtitle summarizes the article as "Researchers are on the trail of a mysterious connection between number theory, algebra and string theory" and it is a balanced and poetic overview of the history of moonshine, its shadowy generalization, and some recent results in the subfield.

In his 1975 paper, Andrew Ogg actually promised a bottle of Jack Daniel's whiskey to the person who proves the connection (page 7-07: "Une bouteille de Jack Daniels est offerte à celui qui expliquera cette coïncidence"). But this bottle seems more pedagogic, especially for readers who are teenagers. Ogg was tempted to buy the bottle to Fields Medal winner Borcherds but Conway said "no, Borcherds only proved things, and not explained the connection". Well, I think that by now, the connection has also been "explained" (and Conway only disagrees because he thinks that the bottle will keep on motivating an army of bright mathematicians on further work) but it seems that Ogg hasn't given the bottle to anyone yet!

I want to offer you some more technical remarks about these amazing mathematical structures and their new organization. The mathematical structure I want to focus on is the Mathieu \(M_{24}\) group and the error-correcting code, the binary Golay code, from which the group may be deduced.

I want to put things in the broadest possible context. There are many complementary ways to do so here. But one global perspective we may take here is the classification of finite simple groups.

It is the work of hundreds of mathematicians who were making their contributions (mostly) during half a century, completed their work about 20 years ago, and the proof of their theorem was originally distributed in 10,000 pages of technical articles in mathematical journals – although some folks have been working on shortening the proof which is undoubtedly possible.

What are the groups, what does it mean to classify them, and what does the resulting classification actually look like? Well, a group is a mathematical structure that is used to describe symmetries mathematically. Technically, it is the set \(G\) of elements – you may interpret each element as an operation \(g\) – that leave some "object" fixed.

But you may (and perhaps you should) forget about any particular visualization of the object. The only thing that you remember is the multiplication table \(gh\) for all the elements. The multiplication is nothing else than the application of both operations in a row; \(gh\) means that you first transform the object by \(h\) and then by \(g\). This order is a convention – and I chose the convention in which the transformed object may be the pure vector \(\ket\psi\) in a quantum mechanical model. It is always possible to imagine that the objects are "wave functions" and the operations themselves (the elements of the group) are linear operators and/or matrices.

Just like the matrix multiplication, the product (composition; the [meta] "operation" combining two operations in the group) must always be associative, \((ab)c=a(bc)\), so that only the ordering of the operations (in this case \(c\), then \(b\), then \(a\)) matters. One of the elements is the identity element \(g=1\) that doesn't change the object at all, so that \(1h=h1=h\) for each \(h\), and there is an inverse element \(g^{-1}\) for each \(g\) such that \(gg^{-1}=g^{-1}g=1\). A group doesn't have to be commutative, \(ab\neq ba\), and most groups are not. But if this holds universally, we call it a commuting or Abelian group (and in that case, we may use the addition \({+}\) as a symbol instead of the multiplication \({\cdot}\) and call it the additive, not multiplicative group).

Well, faces are not exactly \(\ZZ_2\)-symmetric but pretty women get pretty close. Well, the "mirrored right sides" Florence Colgate seems fatter and different but it's OK.

It's an extremely natural definition and the groups – sets of objects like that – are important everywhere. You may always ask what is the symmetry group of some real-world or mathematical object, what is the group of transformations that preserve some "structure" or "characteristic features" of the object. A face is approximately \(\ZZ_2\) symmetric – it's an Abelian group that you may visualize as the set \(\{+1,-1\}\) with the usual multiplication. But there are many more interesting groups.

The classification – the way how to list all the groups – has been completed for finite simple groups. They're finite if the sets contain a finite number of elements – the number of elements is a finite integer. So they're obviously "discrete" groups. Moreover, if you repeat any operation many times, you get back to the identity.

The word "simple" has a technical meaning. It means that the group contains no "normal" or "invariant" (synonymous) subgroups. A normal subgroup \(H\) of group \(G\) is a set \(H\) such that \(gHg^{-1}=H\) for any \(g\in G\) – where \(gHg^{-1}\) is meant to denote the set of all \(ghg^{-1}\) where \(h\) is taken over all \(h\in H\). That's how Wolfram Mathematica would deal with products involving sets.

You may define a direct product \(G\times H\) of two groups. It's a set of all ordered pairs \((g,h)\) where \(g\in G\), \(h\in H\), and the multiplication (composition) applies to the first, \(G\)-like, and the second, \(H\)-like, component of the pair separately. Unless \(G=\{1\}\) or \(H=\{1\}\) i.e. unless one of the factors is a trivial 1-element group (that only contains the identity), \(G\) as well as \(H\) are normal subgroups of \(G\times H\), and \(G\times H\) is therefore not simple.

However, the direct product is not the only way how you may build a non-simple group. You may also define the semidirect product \(G\rtimes_\phi H\) in which the first component of the product isn't \(g_1g_2\) but \(g_1\phi_{h_2}(g_2)\) i.e. twisted by some homomorphism. In that case, \(G\) is still a normal subgroup although the whole group isn't a direct product.

As you can see, non-simple groups are "almost" direct products of simple groups, or semidirect ones, and their numbers of elements are simply the products of the numbers of elements of the factors. In this sense, the whole production of non-simple groups is analogous to getting composite numbers out of primes – except that primes may only be multiplied while the simple groups may be "directly" or "semidirectly" (many types classified by homomorphisms) multiplied.

The list of finite simple groups is therefore the group theory's counterpart of the list of primes, \(2,3,5,7,\dots\).

Now, what the list is? What is the result of the classification?

Even though the proof required 10,000 complicated pages, the sketch of the result seems to be concise. The big theorem says that a finite simple (="prime") group has to be (isomorphic to i.e. being a relabeled copy of) either [read only the bold face if the blockquote below looks too long to you]
  • a cyclic group \(\ZZ_p\) which is the additive group \(\{0,1,2,\dots p-1\}\) where the addition occurs "modulo \(p\)" and where the number of elements \(p\) (the "order" of the group) is a prime for the group to be simple. Note that they're Abelian groups. You may also represent \(\ZZ_p\) as the group of rotations around a point by multiples of \(360^\circ/p\); a subgroup of \(U(1)\) i.e. absolute-value-one complex numbers whose multiplication gives you these rotations. Every finite Abelian group is a direct product of groups \(\ZZ_{p^n}\) whose order is a power of prime, but you need a strict prime for the simplicity.
  • an alternating group \(A_n\). The simplicity condition requires \(n\geq 5\); this condition is related to the fact that 5th and higher order algebraic equations can't be analytically solved. The alternating group \(A_n\) contains all even permutations of \(n\) elements (i.e. permutations that may be written as a product of an even number of transpositions); so its order is \(n!/2\).
  • a simple group of Lie type. You may write down lots of finite groups by using the terminology and logic from Lie groups of matrices, \(PSL,U,Sp,O(n,F)\), except that you don't allow the matrix entries to be real or complex numbers i.e. \(F=\RR\) or \(F=\CC\). Instead, for the sake of finiteness of the group, you choose \(F\) to be a finite field (something like the real numbers or complex numbers, a set where you may add and multiply the elements with the usual conditions), and those are completely classified and it's much easier than to classify the simple groups. You may also replace the "easy" Lie groups by the exceptional ones \(E,F,G\) – yes, those with the Dynkin diagrams but with real/complex numbers replaced by finite fields – and by twisted groups – those with some extra upper left numerical superscript.
  • Semisporadic, Tits group. I wrote it as an extra category because it's sometimes counted in the previous kingdom and sometimes to the following one – as the 27th sporadic group. It's "almost" like one of the groups above \({}^2 F_4(2)'\) derived from the exceptional Lie group \(F_4\) with some twist except that for one particular choice of the twist and the field, you need to make one more step, to consider a "derived subgroup", which is almost the same group as the original one but not quite, and that's why you also lose a "BN pair" so the Tits group doesn't quite agree with the properties in the previous "Lie type" category. You may count it as the easiest – but not smallest – among the "27" sporadic groups. The Tits group wasn't named according to any body parts.
  • Twenty-six sporadic groups. At the level of humanities, the word "sporadic" means almost the same as "exceptional" but the latter word had already been taken so a new adjective was reserved for groups that are "exceptional" in a completely new way. As I have mentioned, the Tits group is sometimes counted as the "most similar to the regular ones", 27th sporadic group. Its order is less than 18 million but other sporadic groups are smaller. On the contrary, the largest sporadic group is the monster group followed by the baby monster group. The Mathieu \(M_{24}\) group is the largest Mathieu sporadic group and I will call it the third most fundamental sporadic group.
That's it. You may see that the "amount of wisdom" (or text needed to capture it) is finite but the ecosystem of the finite groups is very diverse, anyway. Like the "evolutionary tree of life", it contains lots of animals of different complexity with various relationships to other things, variable number of subspecies (breeds and races), and so on.

It's really fascinating – but quite typical in deep mathematics – that a simple task such as "classify all finite simple groups" (with simple and natural definitions) leads to a similarly complex, structured answer.

In the list of the "kingdoms of groups" in the classification above, the complexity is increasing as you go from the top to the bottom. Well, the complexity is a subjective matter (not a rigorously defined one) but I think that almost all mathematicians would agree. As I mention at the end, it may only be humans who see it in this way; God may see it in the opposite way. ;-)

Rubik's cube example

To show you a complex enough example – a good one to see how it typically works – let's consider the Rubik's cube group (Flash). One may perform operations with Rubik's cube and they form a finite group. The operations may be mapped, in a one-to-one way, to different states of the cube (because you may get any state of the cube by an "operation" from the chosen benchmark state, e.g. the sorted one).

It's a cute toy, Mr Rubik has earned tons of money, and everything seems to fit together. So if I ask you where the Rubik's cube group fits into the classification of the finite groups, you may be tempted to say that it's somewhere near the end. Perhaps, it's the monster group, isn't it?
Off-topic: Leslie Winkle and her equally subpar loop quantum gravity colleagues were proved wrong once again. Photons move by the same frequency-independent speeds and only the kind of "quantum foam" that can't change this fact is allowed.
However, despite all the money, the finite group of the operations with Rubik's cube is one of the early boring ones. Well, first of all, the group isn't simple. In other words, it has a normal subgroup. What is it? It's the group \(C_0\) of all the operations that preserve the location of every block but may rotate the blocks around (the corner blocks by \(\ZZ_3\); the middle-of-the-edge blocks by \(\ZZ_2\)). It's a normal group because if you conjugate such a "local rotation of the blocks" by a permutation of the blocks, the permutations cancel and you will get some, generally different "local rotation of the blocks" again (it's generally different because we are \(\ZZ_2\) or \(\ZZ_3\) rotating different, permuted blocks than without the conjugation).

This group \(C_0\) is actually totally boring. It is isomorphic to the Abelian group\[

C_0 = \ZZ_3^7 \times \ZZ_2^{11}.

\] The cube has 8 corners and almost all of them, except for one, may be rotated by \(\ZZ_3\) freely. However, the "required" rotation of the last 8th corner is determined by the other seven; recall that if only one corner is rotated by 120 degrees, there is no way to fix it. That's why the exponent with \(\ZZ_3\) is just seven. And similarly, the cube has twelve edges but one last middle-of-the-edge wrong block can't be fixed, so the exponent above \(\ZZ_2\) is just eleven.

Great. So the normal group is totally boring. The Rubik's cube group is a semidirect product\[

G = C_0 \rtimes C_p

\] where I have to describe the other factor of the semidirect product, \(C_p\). I should also describe the homomorphism needed to construct the semidirect product but it wouldn't be too difficult. The group \(C_p\) itself is actually also unremarkable,\[

C_p = (A_8\times A_{12}) \rtimes \ZZ_2.

\] It's just "all the even permutations of the 8 corner blocks" and "all the even permutations of the 12 middle-of-edge blocks" [thanks for the fix], and some extra \(\ZZ_2\) operation that mixes them in a correlated way; I actually think that the extra \(\ZZ_2\) simply means that an odd permutation of the corners is allowed in combination with an odd permutation of the edges. (It's quite common that only even permutations may be obtained; it is also the case of Loyd's 15 puzzle.) The homomorphisms needed for the two semidirect products contained in \(G\) and \(C_p\) respectively deserve some extra discussion but if you look at the "simple factors" that appear in the Rubik's cube group, they are just \(\ZZ_2,\ZZ_3,A_8,A_{12}\), and that's it. The group theory of Rubik's cube is just a simple conglomerate of several simplest simple finite groups in the classification.

Off-topic: create a 2-minute video showing why the LHC rocks and win a contest organized by the Fermilab, an Illinois-based fan club of the LHC. If the physicists and P.R. folks pick your work (free of obscenities) – sent before the end of May – you will get tickets for 2 from the U.S. to Chicago plus a visit to the Fermilab.

No groups of the Lie type and no sporadic groups are involved at all! And no Tits, either. If you want to become able to "solve" the cube, you must identify the "elementary moves" i.e. the rotations of the faces/layers as products of the "mathematically elementary" generators of the \(\ZZ_2,\ZZ_3,A_8,A_{12}\) factors, roughly speaking, and revert this relationship so that you will be able to permute the individual blocks (by a sequence of rotations of the faces) and then rotate them "almost separately".

Note that the usual "algorithms for the mortals" first tell you how to place the blocks of the "upper, first layer" at the right places; then how to locally rotate them if needed. The sequences of moves are rather simple and you have lots of freedom because you are allowed to bring new disorder to the "second layer" and the "third layer". Then you do the same thing to the second layer – first doing the sequences of moves that bring the blocks to the right places; and finally rotate them to the right orientations if needed. You are allowed to bring new chaos to the third layer but not to the first layer that has already been polished. Finally, you need to sort the "bottom, third layer". The sequences to do it are longer because you have to preserve the first two layers. Again, you first settle the right locations of the blocks, and then rotate them if needed. I already told you that if one corner block or one middle-of-edge block remains rotated, it's because somebody has dismantled the Hungarian cube before you, and you must break it again to fix the problem (or peel the stickers).

Looking at the sporadic groups

The groups of the Lie type are more complex – especially those based on the exceptional Lie groups – but the sporadic groups are even more remarkable. They're 26 or 27 unusual beasts – beasts that, unlike all other simple finite groups, don't arrive in infinite families. They cannot be "mass produced", if you wish. They depend on no integer parameters that could be made arbitrarily large.

Each sporadic group requires a special discussion and boasts its individual virtues and problems. I have said that the semisporadic, Tits group, arises from some technical problem that appears when the exceptional Lie group \(F_4\) twisted in an allowed way is using one of the finite fields based on \(\ZZ_2\).

On the contrary, the largest sporadic groups have much more grandiose stories. The largest sporadic group is the monster group – with almost \(10^{54}\) elements. This is the group related to the \(j\)-function, in some sense the "most important" function on the fundamental domain of \(SL(2,\ZZ)\). The monster group is the largest sporadic group and the master among them – in a similar way in which \(E_8\) is the daddy of exceptional Lie groups. But I need to emphasize that the monster group is in no way "the same thing" as \(E_8\). Their mathematics is equally different; they just happen to be the tips of two icebergs.

The first TRF text about the monstrous moonshine was written in 2006 and many others were added later. String theory has explained why numbers like \(1+196,883\) appear at two seemingly totally unrelated places: one may construct a perturbative string theory, a conformal field theory on the world sheet (some bosonic string compactified on the 24-dimensional torus derived from the Leech lattice, roughly speaking; the Leech lattice is the unique 24-dimensional even self-dual lattice without the vectors of the minimum length that lattices \(\Gamma^{16}\) and \(\Gamma^8+\Gamma^8\) have to produce the \(SO(32)\) and \(E_8\times E_8\) gauge groups), and show that its spectrum enjoys the monster group symmetry. The degeneracies of the states must therefore be (easy enough) dimensions of representations of the monster group, and so on.

Witten has brought evidence that this CFT (with some boundary conditions and co-existence of the left-movers and right-movers) is the holographic AdS/CFT dual of the pure gravity just with black holes in \(AdS_3\), in some sense one of the most structureless theories of quantum gravity. No local graviton or matter excitations there (it's 2+1D), just black holes. It's remarkable – one of the seemingly "most boring" theories of quantum gravity actually has the "most fascinating and largest sporadic" discrete group of symmetries if converted to the exact CFT description.

(Gaiotto showed that only the "minimum" radius comparable to the Planck length has a chance to work – the infinite family of increasingly flat \(AdS_3\) spaces don't admit the monstrous \(CFT_2\) description conjectured by Witten.)

While this complementarity between "super simplicity" and "super complexity" seems intriguing and arguably a principle of mathematics and Nature, I have very limited intuition for "why" the monster group exists at all and how I should imagine it in a way that "fits my brain" completely. The second largest sporadic group is the baby monster group. Its order is over \(10^{33}\), more than the square root of the order of the monster group, and it may be defined as a centralizer of a \(\ZZ_2\) subgroup of the monster group (and probably in many other ways that are harder to formulate).

Mathieu group, umbral moonshine, K3 surfaces, Golay code

I want to spend much more time with the group \(M_{24}\), the largest one among the Mathieu sporadic groups, and related mathematical structures. It is the third most fundamental (but not the third largest, according to the order) sporadic group after the monster and baby monster but I am sure that mathematicians would already disagree at this point (in both ways: John Conway – who is arguably the history's most important explorer of sporadic groups – actually considers \(M_{24}\) to be the most amazing finite group in all of mathematics; I think that K3 linked to this group is cool but, in some sense, "the second" in its depth after the tori and similar simple things).

All the interesting observations below are related to this group. It is sensible to imagine that the amount of "stunning mathematics" of a similar kind is approximately 26 times larger than what you see below, and all of it is "qualitatively different".

First, what is the \(M_{24}\) group? It is a group with almost 245 million elements, substantially fewer than the monster group or the baby monster group. You may build it in various ways from "more regular" groups such as \(PSL(3,4)\). But if you want to see a "full object" whose symmetry group is \(M_{24}\), you can have it: it is the binary Golay code.

Blue is zero, red is one.

The picture above describes "all the nontrivial information" that you can't easily remember and that is needed for the construction of the code. The \(M_{24}\) group – along with the K3-related quantities within string theory – "automatically follow" from this structure if you study it well enough.

The Golay code was discovered in a remarkable 1949 paper by Marcel Golay. Tons of wisdom are linked to this unusual mathematical structure but the original paper – see it by clicking at the URL in this paragraph – was just half a page long! Moreover, Marcel Golay was a guy working for Signal Corps Engineering Laboratories in New Jersey. Those who say that research labs in commercial companies can't produce valuable pure science may look at yet another disproof of their assertion.

What is the problem that Golay was solving? He played with noisy transmission of information, Shannon entropy (greetings to Shannon), and so on. The general problem is how to efficiently transmit (let's only consider binary) information if there is some risk that several of the bits will be reverted due to noise.

Imagine you want to transmit 12 bits of information. If you just transfer 12 bits and some of them are wrong, too bad: the information is transmitted incorrectly. You may transfer those 12 bits twice. If the first copy of the 12 bits disagrees with the second copy due to an error, you know that there is an error. But you don't know which of the two copies of the 12-bit word is right. Moreover, if there are 2 wrong bits among the 24 bits, it may happen that the errors appear on the same place, and you won't even recognize that the seemingly legitimate information (two identical groups of 12 bits) is corrupt. Sending the 12 bits thrice is better – at least, you may pick the "majority form of the 12-bit word" among the 3 copies, with a higher chance of being right. But if there are 2 erroneous bits, it may still happen that you will send corrupt information that looks right – despite the tripled number of information you have sent.

There are better ways to transmit bits so that you may fix the errors and/or be sure that the result is OK, assuming that the percentage of errors remains relatively low (but it's allowed to be higher than 1 wrong bit). And the binary Golay code is one of the greatest – and, in fact, also most important in practice – error-correcting codes.

Let me post this diagram again. You see that the left half of the picture contains sequences from 100000000000 up to 000000000001 – with 11 zeroes and 1 digit one. For each of these "elementary bits", there exists a 12-bit codeword to check the validity of everything that is written, or fixed a few mistakes. By now, most TRF readers probably know how to use the picture above to send the 12 bits reliably.

Twelve zeroes are sent as twenty-four zeroes. To mention a more difficult example, the first row tells you that instead of 100000000000, you send 24 bits 100000000000100111110001. Other lines tell you how to "encode" other sequences of 12 bits where the digit 1 appears exactly once. If you want to send more complicated sequences of 12 bits, you add the corresponding rows by "EXOR" i.e. modulo two in each column. For example, if you want to send 110000000000, you send "the first row EXOR the second row" which means 110000000000110100001011. You surely know how to encode the most general 12 bits (i.e. all of the 4,096 twelve-bit words), too.

Why is exactly this choice of the 12 extra verification bits special? It's because each two allowed sequences of 24 bits differ at many places – they differ by 8 or more bits (among the 24). This (minimum) number of "different bits" among two allowed code words is known as the (minimum) Hamming distance. For the binary Golay code, it happens to be 8 which is a lot. If you "damage" a few random places in the table, the distance will probably be smaller than 8; the "damaged" algorithm to transmit information will be less reliable than Golay's correct one.

You are invited to verify that the distance is never less than 8 on a few examples. For example, pick two random rows in the table expressed as the image and count the number of bits (among the 24) by which the two rows differ. They differ by 2 bits among the first 12 (the \(i\)-th and \(j\)-th bit, of course), and the difference in the remaining "chaotic" 12 bits will always be in 6 (or at least 6) bits. It just works. You need to check the distances for all pairs taken from the 4096 allowed words, not just pairs taken from the 12 "generators", however.

As long as there are at most 7 erroneous bits among the 24, you may safely recognize the "correct" and "damaged" sequences of the 24 bits: there is no risk that the errors will actually create another allowed 24-bit codeword. Moreover, if at most 3 bits among the 24 bits are erroneous, you will know how to fix them. You may be somewhere on the length-8 path between two allowed codewords but to be in the middle, you must be 4 erroneous bits from either side, so if the number of erroneous bits is at most 3, you know "where you should go".

If you use this trick purely for error correction, you may say that it would be enough for the minimum Hamming distance to be 7, and not 8, because if the distance between two allowed codewords is 7, you may still have 3 errors and you know into which side you should move because 3 is still less than 7/2. You may achieve the minimum Hamming distance 7 if you just drop 1 of the 24 bits. The corresponding code is the 23-bit "perfect binary Golay code" \(G_{23}\). This reduction may be helpful for the information science application but as far as I can say, it makes the mathematical structure less natural from the viewpoint of fundamental mathematics and physics which is why I will always talk about the 24-bit code in physics-related texts (and below). It's the more natural mathematical structure, despite the missing adjective "perfect".

It's a cool code which may be very useful in actual transmission of signals in noisy environments. But it has far-reaching implications for mathematics and physics – via string theory.

The reason is that the "automorphism group of the binary Golay code" is the Mathieu group \(M_{24}\) (and it is similarly the less cool and smaller sporadic group, \(M_{23}\), for the truncated "perfect" 23-bit code; if you want to know, Émile Léonard Mathieu introduced the first known sporadic groups \(M_{11},M_{12},M_{22},M_{23},M_{24}\) in 1861 and 1873 and the subscripts are really numbers between eleven and twenty-four although they look like pairs of small integers). One may define the group as a subgroup of the \(S_{24}\), the permutation of the full 24 bits in the code, that leaves the set of \(2^{12}=4096\) allowed codewords "the same" as a set.

(It's historically remarkable that pretty much a century of silence came after the discovery of the Mathieu groups and the following sporadic group, J1, was described by Zvonimir Janko only in 1965. Only afterwards, things sped up: 21 sporadic groups were found within an M-theory-extended decade between 1965 and 1976.)

So a particularly clever and special error-correcting code described in a half-a-page paper from 1949 (and in group-theoretical papers that were much longer but almost 100 years older) is enough to define the third most fundamental sporadic group in group theory! The Mathieu \(M_{24}\) group is the symmetry group of the code.

I must also mention that the binary Golay code (and therefore the group) is also closely related to the Leech lattice (that may be used to define the CFT with the monster group). Why? Because in the Leech lattice, allowed coordinates modulo 8 (times the "quantum" of the coordinate) are in one-to-one correspondence with the allowed binary Golay codes. To get from the monster group, you first need to throw away all the "purely stringy" elements of the symmetry group (only keep the symmetries of the Leech lattice), and then pick those that are a subgroup of \(S_{24}\).

Your screen doesn't have sufficiently many dimensions but this "animated quartic" conveys the spirit of what the K3 surfaces look like.

Relationship to K3 surfaces

If you consider the "simplest", most (super)symmetric compactifications of string/M-theory, the toroidal compactifications are the first ones you consider. They preserve all the supersymmetries of the decompactified spacetime. The simplest non-flat compactification manifold is the K3 surface, one of the family of 4-real-dimensional curved manifolds, a 4-real-dimensional counterpart of the "Calabi-Yau manifolds" (TRF random search, Aspinwall's introduction). One-half of the supersymmetry is preserved. M-theory or type II string theory on a K3 surface is dual (equivalent) to heterotic string theory on a torus (it's called the string-string duality).

On the world sheet, you may calculate some kind of a twisted partition function of the conformal field theory that describes strings on a K3 manifold perturbatively. This partition function is known as the "elliptic genus"\[

Z_{ell}(\tau;z) = {\rm Tr}_{{\mathcal R}\times {\mathcal R}} (-1)^{F_L+F_R} q^{L_0} \bar q^{\bar L_0} e^{4\pi i z J_{0,L}^3}

\] where the extra exponential twists the partition sum by a transformation in the affine \(SU(2)\) algebra – that is another local gauge symmetry on the world sheet just like the diffeomorphisms and Weyl symmetry (plus the local world sheet supersymmetry). The power of \((-1)\) turns this partition function into a supertrace, not a trace, so there are lots of cancellations – similar to those that appear in the indices (the singular is "index"). However, this cancellation is only partial which is why the elliptic genus effectively gets contributions from some "short representations" of SUSY but it remembers some of the properties of these representations. The elliptic genus is therefore kind of holomorphic and "something in between" the index, which is a simple integer, and the generic partition sum, which is a non-holomorphic function.

(If you care about a general technicality explaining something from the previous sentence, note that the exponential factor involving \(J_{0,L}^3\) in the supertrace depends on \(L\), the left-movers only, and breaks the symmetry between the left-movers and right-movers on the world sheet. Due to this extra factor which is a sign, the representations that are completely annihilated by the right-moving supersymmetries see a complete Bose-Fermi cancellation in the supertrace; while those annihilated by the left-moving supercharges don't. [Or vice versa? Be careful if you need that.] That's why the elliptic genus behaves as an index from the viewpoint of the right-moving degrees of freedom; but as a partition sum from the viewpoint of the left-moving ones. The elliptic genus is literally a heterosis – a left-right hybrid – of an index and a partition sum, in the same sense in which the heterotic string is a heterosis of the bosonic string and the superstring. And that's why it, the elliptic genus, has a holomorphic dependence but no anti-holomorphic one.)

In 2010, Eguchi, Ooguri, and Tachikawa noticed that there apparently exists a new kind of a moonshine that involves the perturbative string theories involving the K3 surfaces (TRF 2010).

In the ordinary monstrous moonshine, one expands the \(j\)-function as\[

j(\tau) = \frac 1q + 744 + 196884 q + 21493760 q^2 + \dots

\] where \(q\equiv \exp(2\pi i \tau)\) and sees dimensions of simple representations of the monster group such as \(1+196,883\) everywhere. Similarly, Eguchi et al. expanded the elliptic genus for K3 – well, I will write the expansion for \(\Sigma(\tau)\) which is related to \(Z_{ell}(K3)(\tau; z)\) by a rather simple relationship that nevertheless depends on modular functions that not everyone knows (equation 1.7 in Eguchi et al.) – and they saw that it was\[

\frac{\Sigma(\tau)}{-2q^{-1/8}} &= 1 - 45 q - 231 q^2 -770 q^3 - \\
&- 2277q^4 -5796 q^5 -\\
&-13915 q^6 - 30843 q^7-\dots

\] Much like in the case of the \(j\)-function, the coefficients in front of \(q^n\) are rather interesting integers. Well, they are integers smaller than and less impressive than 196,884. But there are many of them that are interesting enough.

Well, the fun is that if you look at dimensions of irreducible representations of the Mathieu \(M_{24}\) groups, you will find numbers \(1,45,231,770,2277,5796\) among them (among just 20 similarly large numbers describing the dimensions of the irreps). The following two coefficients, \(13915,30843\), are dimensions of reducible representation, i.e. simple and unique sums of (two or six) numbers describing the dimensions of the irreps. The following coefficients (not shown above) may also be decomposed but the decomposition is no longer unique.

Because of the experience with the monstrous moonshine, Eguchi et al. were already pretty much sure that the agreement between these numbers can't be a coincidence. There must exist an explanation – a different one than in the case of the monstrous moonshine; but one that plays the same role – which links the two seemingly different mathematical tasks, namely the third most fundamental sporadic group with the partition functions on K3 surfaces in string theory.

The task of demystifying these connections becomes virtually complete when one constructs the corresponding perturbative string theory with the sporadic symmetry; but with demonstrable links of its (twisted) partition sums to the modular functions such as \(j\) and \(\Sigma(K3)\) above. Mathematicians are spending lots of time and they have proved "almost everything" that satisfies them – which is not quite the complete "visualization in terms of string theory" but it is close.

They like to prove that it is possible to write the modular functions as some series – McKay-Thompson series – of some infinite-dimensional representations of a certain type. The "infinite-dimensional representations" (or "graded modules", in the refined jargon of the mathematicians), are "almost" the spectra of the relevant string theory, but they don't construct the string theory as explicitly as string theorists would want so I think it's fair to say that the mathematicians as not cracking the problem as completely as physicists (string theorists) would demand.

So far, the latest proof in this mathematical industry was published in early March 2015. Duncan, Griffin, and Ono have developed the proof (linking the modular forms with some infinite-dimensional representations via some series) in the case of 22 remaining examples of "umbral moonshine" that were conjectured in the literature. The proof for \(M_{24}\) that we focused upon was settled by Terry Gannon in late 2012 (while some previous insights were made by Gaberdiel et al.). You may see that the progress is relatively fast here.

Here, things get very technical – and they're not formulated in the physicist-friendly language I would find easy to devour – so let me remain superficial. The adjective "umbral" is derived from the Latin word "umbra" for a "shadow" – and it's used for these non-monstrous versions of moonshine because the corresponding mock modular forms always allow you to compute an affiliated modular function that is a "shadow" of the mock modular form. The "umbral moonshine" theorem is meant to be a generalization of the "Mathieu moonshine" to numerous other groups that, like \(M_{24}\), urge you to use the shadow modular functions; the monstrous moonshine doesn't belong here.

There seems to be an intermediate step which may be the reason why it is believed that the string theories unmasking these types of moonshine don't actually have the exact sporadic symmetry group whose representations appear; but they have a symmetry group related to it in some way, too.

Some complications exist but at the end, I believe that sometime in the future, people will have the full description of the relevant string theories that unmask all the shocking surprises. Maybe all of them will be some rather simple orbifolds etc. based on the Leech-lattice CFT we know from monstrous moonshine. I find it likely that the "24 bits" the Golay code will be in one-to-one correspondence with the 24 dimensions in the Leech lattice – effectively with the 24 purely transverse dimensions of the bosonic string theory that is helpful. And whenever K3 will be involved, I think that these 24 directions will be mapped to the K3 cohomology, so the 24-dimensional flat description will be closely related to the heterotic dual description of a K3 in the string-string duality, despite the wrong 24+0 (in the \(M_{24}\) structures) instead of 20+4 signature (intersection numbers of homologies in K3). Note that in the heterotic description, the signature means that the (20) positive-signature dimensions become left-movers and the (4) negative-signature ones become right-movers so this "asymmetry" has to be liquidated in some way and everything must be made left-moving.

Religious implications

This title is perhaps somewhat over the edge – but I think that not too much. What do I mean that these things have religious implications?

I think that when you look at some mathematical structures – or features or compactifications of string/M-theory – they are often ordered hierarchically in a way that resembles the list in the classification of the finite simple groups.

Note that at the beginning, you have the "easy" structures that may be constructed from pieces that are available to humans. \(\ZZ_p\) and then \(A_n\) and slightly more complicated ones. Things get more complicated but when clumped properly, the path to the most shocking and exceptional structures – ending with the sporadic groups and the monster group in particular – is finite.

My meme is that the easy constructive groups like \(\ZZ_2\) are close to humans who are mortal; low-brow individuals such as Lee Smolin proudly remain attached to this wild primitive animalistic side of the chain. And the opposite side is close to God (and refined string theorists). OK, I hope you forgave me that religious interpretation. ;-)

We understand things that are human but there is a dual perspective of God who primarily understands the things on the opposite end, like the monster group, and who needs to perform some special activity – i.e. offer the apples that explain sex – in order to introduce sins and to break the beauty of the divine world and to create all the mundane unspiritual stuff that we know from the everyday life, like the Rubik's cubes I described in some detail.

The path between Man and God is finite, when organized properly, and the more we internalize the thinking that makes the sporadic groups or the monster look like the easy starting points, the closer to the perspective of God we become. All truly valuable ideas in mathematics and physics may perhaps be organized using this divine, string-theory-based perspective on a sunny day in the future. Even the realistic vacua of string/M-theory will be viewed as a partial symmetry breaking of the truly God-like compactifications boasting things like the monster group symmetry. Monstrous insights will therefore be a part of the answer to currently controversial questions such as "what was there before the big bang".

And that's the memo.

The first bonus: the hierarchy of power organizing the sporadic groups. It's remarkable that such a messy graph is "pure mathematics", isn't it? Extraterrestrials may draw it in the same way.

The lines essentially denote embedding as subgroups. The monster M is at the top, and the baby monster B and 18 other sporadic groups consider the monster M their holy father; there are 20 members of this Monstrous Catholic Church. (The Catholics call themselves "the happy family" which is clearly propaganda, so I prefer "Catholics" LOL.) There is some opposition, too. The six groups away from the monster-led hierarchy are not known as heretics or renegades or mavericks. It would be too silly to use such words. Instead, they are the pariah groups. ;-)

They include J1 (a group that may be uniquely specified by some 2-Sylow subgroups and their properties), J3, J4, the three of the J-groups – Catholic J2, the Hall-Janko group, is mostly linked to these three sociologically, by the Croatian mathematician Zvonimír Janko who found the four, not by some intrinsic system. You see that J4, a pariah, was rather close to the church, anyway, and \(M_{24}\) emphasized in this article is a secret child of the Monstrous Pope M and the pariah J4. That could be related to the fact that this is linked to K3 surfaces which are dual to the heterotic string, also a hybrid of two very different parents. In the right lower corner, you see J1 embedded in the O'Nan group ON, and J3 along with the Rudvalis group Ru.

The last pariah I haven't mentioned is the (far left) Lyons group Ly. Like other far leftists, the Lyons group seems to have no importance besides its existence. I hope that the Catholic readers will be thrilled to learn that to a lesser extent, it is the case of most pariahs. Well, so far. One wants to believe that the other sporadic groups must also be "comparably important" to M or \(M_{24}\) and we're only ignorant about their role because the pariahs have been discriminated against; but the alternative assumption that they're really useless junk is plausible, too. ;-)

P.S. Looking at the moonshine things may be a great way to spend the remaining 315 days that we have before the planet becomes a total frying pan according to the world's president of the climate alarm. ObamaCare isn't sufficiently funded to store this individual in a psychiatric asylum yet.

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snail feedback (18) :

reader RAF III said...

Lubos - Wonderful post!!! How do you find the time to write as much as you do? Are you super fast, like the Flash, or what?

reader Luboš Motl said...

Thanks, RAF! This actually did take something close to 4 hours up to the moment when I proofread it. It is not as straightforward as to write a blog post about a war, for example. ;-) Your Flash idea is a good one, I am going to learn this:


Oops, my sister just complains that her new notebook is showing her "Flash plugin is not responding" in recent days. Well, not everyone may be OK with using the power of Flash. ;-)

reader RAF III said...


reader Liam said...

Hey funny coincidence - a Hungarian friend gave me a Rubik's cube for my birthday. He complained that I hadn't touched it and scrambled it for me, so I looked up on the internet the beginner row-by-row method you describe but inverted the colors of the edge pieces to make nice yellow/white, green/blue, orange/red checkerboard patterns on the faces, LOL :)

People find it strangely impressive, and I'm always accused of having cheated with the stickers :-P

This looks like a great review post - informative but accesible.. I look forward to digesting it this evening!

reader kneemo said...

M24 indeed has a simple interpretation as part of the symmetry of 24 mutually orthogonal vectors in the 829,2375 (Z2 mod) crosses of 48 vectors of norm 8. Each such cross has the symmetry of the Golay code, with order 2^12 × |M24|. As each cross can be taken as the coordinate system of the Leech lattice, we have:

|Aut(Leech)|=|Co_0|= 829,2375 x 2^12 × |M24|.

reader Tom Hendrix said...

"all the even permutations of the 12 corner blocks" and "all the even permutations of the 8 middle-of-edge blocks"

Are 12 and 8 backward?

reader Leo Vuyk said...

Dear Lubos,

How would you classify a system of Tori deratives as
I propose at the attachment poster?

reader Uncle Al said...

String theory's deep broad mathematical roots are unquestionable and beautiful. String theory is not empirical. Remove pieces that cannot reduce to practice.

Are vacuum symmetries toward massless boson photons exactly those toward massed fermion quarks? Highly unlikely but postulated, then excuses. The universe contains no visible Equivalence Principle violation. Chemistry does things that physics cannot imagine. Look, DOI: 10.5281/zenodo.15107.

(A liquor-based bet should meet or exceed Lagavulin 16 single malt scotch.)

reader RAF III said...

Try Ardbeg before anyone accepts your bet. You might want to change the terms.

reader Uncle Al said...

A jigger of Laphroaig (Ardbeg) over very cold French vanilla ice cream. Lagavulin sets a good baseline; exceed at will.

OTOH, Canadian whiskey is whiskey-flavored vodka and Irish whiskey is whiskey-flavored Pepsi less the carbonation.

reader RAF III said...


reader DEF conjecture said...

So is this a deep thing for string theory? Are the exceptional groups preferred for reality?

reader Luboš Motl said...

This is a deep thing for everyone who has a brain, not just "for string theory".

Exceptional groups are probably preferred for reality, too. But this blog post wasn't about exceptional groups but about sporadic groups which is something very different.

reader Leo Vuyk said...

If our material universe has at least one symmetric anti-material entangled mirror copy universe, ( Max Tegmark: “Is
there a copy of you reading this article” ) then parity violation seems to be a must in each universe.
Both Higgs fields should have a preferred opposite spiral rotation effect.
The Higgs field oscillations should perform left- to right handed spirals.
See: the bouncing multiverse and the beginning and end of time

reader Luboš Motl said...

Oh I see, it is a raspberry universe. I thought it was the stargate from the Contact 1997 film


reader Liam said...

:D :D

reader RAF III said...

Lubos - Don't worry, I'm not looking to continue our dispute (I assume we have agreed to disagree), but I need a question answered from a halfers perspective. Since I don't get out much and you're the only halfer I know, I thought I would presume on you to answer.
Would it be fair to say that, upon awakening, a halfer would assign the probabilities 1/2, 1/4, 1/4 to the epistemic possibilities (Mon, heads), (Mon,tails), (Tue,tails), and then, upon *not* being told the day, update them to P(heads) = P(tails) = 1/2, much as one would in the Monty Hall problem?
You seem to imply as much in a comment below.

reader Leo Vuyk said...

For us humans it is hard to imagine that we are entangled over the edge of our own universe (behind the horizon) and as such could have to deal with our opposite self at very long distance.