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Category: noncommutative

What we (don’t) know

Do we know why the monster exists and why there’s moonshine around it?

The answer depends on whether or not you believe that vertex operator algebras are natural, elegant and inescapable objects.

the monster

Simple groups often arise from symmetries of exceptionally nice mathematical objects.

The smallest of them all, $A_5$, gives us the rotation symmetries of the icosahedron. The next one, Klein’s simple group $L_2(7)$, comes from the Klein quartic.

The smallest sporadic groups, the Mathieu groups, come from Steiner systems, and the Conway groups from the 24-dimensional Leech lattice.

What about the largest sporadic simple, the monster $\mathbb{M}$?

In his paper What is … the monster? Richard Borcherds writes (among other characterisations of $\mathbb{M}$):

“3. It is the automorphism group of the monster vertex algebra. (This is probably the best answer.)”

But, even Borcherds adds:

“Unfortunately none of these definitions is completely satisfactory. At the moment all constructions of the algebraic structures above seem artificial; they are constructed as sums of two or more apparently unrelated spaces, and it takes a lot of effort to define the algebraic structure on the sum of these spaces and to check that the monster acts on the resulting structure.
It is still an open problem to find a really simple and natural construction of the monster vertex algebra.

Here’s 2 minutes of John Conway on the “one thing” he really wants to know before he dies: why the monster group exists.



moonshine

Moonshine started off with McKay’s observation that 196884 (the first coefficient in the normalized j-function) is the sum 1+196883 of the dimensions of the two smallest simple representations of $\mathbb{M}$.

Soon it was realised that every conjugacy class of the monster has a genus zero group (or ‘moonshine group’) associated to it.

Borcherds proved the ‘monstrous moonshine conjectures’ asserting that the associated main modular function of such a group is the character series of the action of the element on the monster vertex algebra.

Here’s Borcherds’ ICM talk in Berlin on this: What is … Moonshine?.

Once again, the monster vertex algebra appears to be the final answer.

However, in characterising the 171 moonshine groups among all possible genus zero groups one has proved that they are all of the form:

(ii) : $(n|h)+e,g,\dots$

In his book Moonshine beyond the Monster, Terry Gannon writes:

“We now understand the significance, in the VOA or CFT framework, of transformations in $SL_2(\mathbb{Z})$, but (ii) emphasises that many modular transformations relevant to Moonshine are more general (called the Atkin-Lehner involutions).
Monstrous moonshine will remain mysterious until we can understand its Atkin-Lehner symmetries.

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the moonshine picture – at last

The monstrous moonshine picture is the subgraph of Conway’s big picture consisting of all lattices needed to describe the 171 moonshine groups.

It consists of:

– exactly 218 vertices (that is, lattices), out of which

– 97 are number-lattices (that is of the form $M$ with $M$ a positive integer), and

– 121 are proper number-like lattices (that is of the form $M \frac{g}{h}$ with $M$ a positive integer, $h$ a divisor of $24$ and $1 \leq g \leq h$ with $(g,h)=1$).

The $97$ number lattices are closed under taking divisors, and the corresponding Hasse diagram has the following shape

Here, number-lattices have the same colour if they have the same local structure in the moonshine picture (that is, have a similar neighbourhood of proper number-like lattices).

There are 7 different types of local behaviour:

The white numbered lattices have no proper number-like neighbours in the picture.

The yellow number lattices (2,10,14,18,22,26,32,34,40,68,80,88,90,112,126,144,180,208 = 2M) have local structure

\[
\xymatrix{M \ar@{-}[r] & \color{yellow}{2M} \ar@{-}[r] & M \frac{1}{2}} \]

which involves all $2$-nd (square) roots of unity centered at the lattice.

The green number lattices (3,15,21,39,57,93,96,120 = 3M) have local structure

\[
\xymatrix{& M \ar@[red]@{-}[d] & \\ M \frac{1}{3} \ar@[red]@{-}[r] & \color{green}{3M} \ar@[red]@{-}[r] & M \frac{2}{3}} \]

which involve all $3$-rd roots of unity centered at the lattice.

The blue number lattices (4,16,20,28,36,44,52,56,72,104 = 4M) have as local structure

\[
\xymatrix{M \frac{1}{2} \ar@{-}[d] & & M \frac{1}{4} \ar@{-}[d] \\
2M \ar@{-}[r] & \color{blue}{4M} \ar@{-}[r] & 2M \frac{1}{2} \ar@{-}[d] \\
M \ar@{-}[u] & & M \frac{3}{4}} \]

and involve the $2$-nd and $4$-th root of unity centered at the lattice.

The purple number lattices (6,30,42,48,60 = 6M) have local structure

\[
\xymatrix{& M \frac{1}{3} \ar@[red]@{-}[d] & 2M \frac{1}{3} & M \frac{1}{6} \ar@[red]@{-}[d] & \\
M \ar@[red]@{-}[r] & 3M \ar@{-}[r] \ar@[red]@{-}[d] & \color{purple}{6M} \ar@{-}[r] \ar@[red]@{-}[u] \ar@[red]@{-}[d] & 3M \frac{1}{2} \ar@[red]@{-}[r] \ar@[red]@{-}[d] & M \frac{5}{6} \\
& M \frac{2}{3} & 2M \frac{2}{3} & M \frac{1}{2} & } \]

and involve all $2$-nd, $3$-rd and $6$-th roots of unity centered at the lattice.

The unique brown number lattice 8 has local structure

\[
\xymatrix{& & 1 \frac{1}{4} \ar@{-}[d] & & 1 \frac{1}{8} \ar@{-}[d] & \\
& 1 \frac{1}{2} \ar@{-}[d] & 2 \frac{1}{2} \ar@{-}[r] \ar@{-}[d] & 1 \frac{3}{4} & 2 \frac{1}{4} \ar@{-}[r] & 1 \frac{5}{8} \\
1 \ar@{-}[r] & 2 \ar@{-}[r] & 4 \ar@{-}[r] & \color{brown}{8} \ar@{-}[r] & 4 \frac{1}{2} \ar@{-}[d] \ar@{-}[u] & \\
& & & 1 \frac{7}{8} \ar@{-}[r] & 2 \frac{3}{4} \ar@{-}[r] & 1 \frac{3}{8}} \]

which involves all $2$-nd, $4$-th and $8$-th roots of unity centered at $8$.

Finally, the local structure for the central red lattices $12,24 = 12M$ is

\[
\xymatrix{
M \frac{1}{12} \ar@[red]@{-}[dr] & M \frac{5}{12} \ar@[red]@{-}[d] & M \frac{3}{4} \ar@[red]@{-}[dl] & & M \frac{1}{6} \ar@[red]@{-}[dr] & M \frac{1}{2} \ar@[red]@{-}[d] & M \frac{5}{6} \ar@[red]@{-}[dl] \\
& 3M \frac{1}{4} \ar@{-}[dr] & 2M \frac{1}{6} \ar@[red]@{-}[d] & 4M \frac{1}{3} \ar@[red]@{-}[d] & 2M \frac{1}{3} \ar@[red]@{-}[d] & 3M \frac{1}{2} \ar@{-}[dl] & \\
& 2M \frac{1}{2} \ar@[red]@{-}[r] & 6M \frac{1}{2} \ar@{-}[dl] \ar@[red]@{-}[d] \ar@{-}[r] & \color{red}{12M} \ar@[red]@{-}[d] \ar@{-}[r] & 6M \ar@[red]@{-}[d] \ar@{-}[dr] \ar@[red]@{-}[r] & 2M & \\
& 3M \frac{3}{4} \ar@[red]@{-}[dl] \ar@[red]@{-}[d] \ar@[red]@{-}[dr] & 2M \frac{5}{6} & 4M \frac{2}{3} & 2M \frac{2}{3} & 3M \ar@[red]@{-}[dl] \ar@[red]@{-}[d] \ar@[red]@{-}[dr] & \\
M \frac{1}{4} & M \frac{7}{12} & M \frac{11}{12} & & M \frac{1}{3} & M \frac{2}{3} & M}
\]

It involves all $2$-nd, $3$-rd, $4$-th, $6$-th and $12$-th roots of unity with center $12M$.

No doubt this will be relevant in connecting moonshine with non-commutative geometry and issues of replicability as in Plazas’ paper Noncommutative Geometry of Groups like $\Gamma_0(N)$.

Another of my pet follow-up projects is to determine whether or not the monster group $\mathbb{M}$ dictates the shape of the moonshine picture.

That is, can one recover the 97 number lattices and their partition in 7 families starting from the set of element orders of $\mathbb{M}$, applying some set of simple rules?

One of these rules will follow from the two equivalent notations for lattices, and the two different sets of roots of unities centered at a given lattice. This will imply that if a number lattice belongs to a given family, certain divisors and multiples of it must belong to related families.

If this works out, it may be a first step towards a possibly new understanding of moonshine.

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a non-commutative Jack Daniels problem

At a seminar at the College de France in 1975, Tits wrote down the order of the monster group

\[
\# \mathbb{M} = 2^{46}.3^{20}.5^9.7^6.11^2.13^3.17·19·23·29·31·41·47·59·71 \]

Andrew Ogg, who attended the talk, noticed that the prime divisors are precisely the primes $p$ for which the characteristic $p$ super-singular $j$-invariants are all defined over $\mathbb{F}_p$.

Here’s Ogg’s paper on this: Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 16, no 1 (1974-1975).

Ogg offered a bottle of Jack Daniels for an explanation of this coincidence.

Even Richard Borcherds didn’t claim the bottle of Jack Daniels, though his proof of the monstrous moonshine conjecture is believed to be the best explanation, at present.

A few years ago, John Duncan and Ken Ono posted a paper “The Jack Daniels Problem”, in which they prove that monstrous moonshine implies that if $p$ is not one of Ogg’s primes it cannot be a divisor of $\# \mathbb{M}$. However, the other implication remains mysterious.

Duncan and Ono say:

“This discussion does not prove that every $p ∈ \text{Ogg}$ divides $\# \mathbb{M}$. It merely explains how the first principles of moonshine suggest this implication. Monstrous moonshine is the proof. Does this then provide a completely satisfactory solution to Ogg’s problem? Maybe or maybe not. Perhaps someone will one day furnish a map from the characteristic $p$ supersingular $j$-invariants to elements of order $p$ where the group structure of $\mathbb{M}$ is apparent.”

I don’t know whether they claimed the bottle, anyway.

But then, what is the non-commutative Jack Daniels Problem?

A footnote on the first page of Conway and Norton’s ‘Monstrous Moonshine’ paper says:

“Very recently, A. Pizer has shown these primes are the only ones that satisfy a certain conjecture of Hecke from 1936 relating modular forms of weight $2$ to quaternion algebra theta-series.”

Pizer’s paper is “A note on a conjecture of Hecke”.

Maybe there’s a connection between monstrous moonshine and the arithmetic of integral quaternion algebras. Some hints:

The commutation relations in the Big Picture are reminiscent of the meta-commutation relations for Hurwitz quaternions, originally due to Conway in his booklet on Quaternions and Octonions.

The fact that the $p$-tree in the Big Picture has valency $p+1$ comes from the fact that the Brauer-Severi of $M_2(\mathbb{F}_p)$ is $\mathbb{P}^1_{\mathbb{F}_p}$. In fact, the Big Picture should be related to the Brauer-Severi scheme of $M_2(\mathbb{Z})$.

Then, there’s Jorge Plazas claiming that Connes-Marcolli’s $GL_2$-system might be related to moonshine.

One of the first things I’ll do when I return is to run to the library and get our copy of Shimura’s ‘Introduction to the arithmetic theory of automorphic functions’.

Btw. the bottle in the title image is not a Jack Daniels but the remains of a bottle of Ricard, because I’m still in the French mountains.

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the monstrous moonshine picture – 2

Time to wrap up my calculations on the moonshine picture, which is the subgraph of Conway’s Big Picture needed to describe all 171 moonshine groups.

No doubt I’ve made mistakes. All corrections are welcome. The starting point is the list of 171 moonshine groups which are in the original Monstrous Moonshine paper.

The backbone is given by the $97$ number lattices, which are closed under taking divisors and were found by looking at all divisors of the numbers $N=n \times h$ for the 171 moonshine groups of the form $N+e,f,\dots$ or $(n|h)+e,f,\dots$.

The Hasse-diagram of this poset (under division) is here (click on the image to get a larger version)

There are seven types of coloured numbers, each corresponding to number-lattices which have the same local structure in the moonshine picture, as in the previous post.

The white numbered lattices have no further edges in the picture.

The yellow number lattices (2,10,14,18,22,26,32,34,40,68,80,88,90,112,126,144,180,208 = 2M) have local structure

\[
\xymatrix{& \color{yellow}{2M} \ar@{-}[r] & M \frac{1}{2}} \]

The green number lattices (3,15,21,39,57,93,96,120 = 3M) have local structure

\[
\xymatrix{M \frac{1}{3} \ar@[red]@{-}[r] & \color{green}{3M} \ar@[red]@{-}[r] & M \frac{2}{3}} \]

The blue number lattices (4,16,20,28,36,44,52,56,72,104 = 4M) have as local structure

\[
\xymatrix{M \frac{1}{2} \ar@{-}[d] & & M \frac{1}{4} \ar@{-}[d] \\
2M \ar@{-}[r] & \color{blue}{4M} \ar@{-}[r] & 2M \frac{1}{2} \ar@{-}[d] \\
& & M \frac{3}{4}} \]

where the leftmost part is redundant as they are already included in the yellow-bit.

The purple number lattices (6,30,42,48,60 = 6M) have local structure

\[
\xymatrix{M \frac{1}{3} \ar@[red]@{-}[d] & 2M \frac{1}{3} & M \frac{1}{6} \ar@[red]@{-}[d] & \\
3M \ar@{-}[r] \ar@[red]@{-}[d] & \color{purple}{6M} \ar@{-}[r] \ar@[red]@{-}[u] \ar@[red]@{-}[d] & 3M \frac{1}{2} \ar@[red]@{-}[r] \ar@[red]@{-}[d] & M \frac{5}{6} \\
M \frac{2}{3} & 2M \frac{2}{3} & M \frac{1}{2} & } \]

where again the lefmost part is redundant, and I forgot to add the central part in the previous post… (updated now).

The unique brown number lattice 8 has local structure

\[
\xymatrix{& & 1 \frac{1}{4} \ar@{-}[d] & & 1 \frac{1}{8} \ar@{-}[d] & \\
& 1 \frac{1}{2} \ar@{-}[d] & 2 \frac{1}{2} \ar@{-}[r] \ar@{-}[d] & 1 \frac{3}{4} & 2 \frac{1}{4} \ar@{-}[r] & 1 \frac{5}{8} \\
1 \ar@{-}[r] & 2 \ar@{-}[r] & 4 \ar@{-}[r] & \color{brown}{8} \ar@{-}[r] & 4 \frac{1}{2} \ar@{-}[d] \ar@{-}[u] & \\
& & & 1 \frac{7}{8} \ar@{-}[r] & 2 \frac{3}{4} \ar@{-}[r] & 1 \frac{3}{8}} \]

The local structure in the two central red number lattices (not surprisingly 12 and 24) looks like the image in the previous post, but I have to add some ‘forgotten’ lattices.

That’ll have to wait…

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Roots of unity and the Big Picture

All lattices in the moonshine picture are number-like, that is of the form $M \frac{g}{h}$ with $M$ a positive integer and $0 \leq g < h$ with $(g,h)=1$. To understand the action of the Bost-Connes algebra on the Big Picture it is sometimes better to view the lattice $M \frac{g}{h}$ as a primitive $h$-th root of unity, centered at $hM$.

The distance from $M$ to any of the lattices $M \frac{g}{h}$ is equal to $2 log(h)$, and the distances from $M$ and $M \frac{g}{h}$ to $hM$ are all equal to $log(h)$.

For a prime value $h$, these $h$ lattices are among the $h+1$ lattices branching off at $hM$ in the $h$-adic tree (the remaining one being $h^2M$).

For general $h$ the situation is more complex. Here’s the picture for $h=6$ with edges in the $2$-adic tree painted blue, those in the $3$-adic tree red.

\[
\xymatrix{& & M \frac{1}{2} \ar@[blue]@{-}[d] & \\
& M \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & 2M \ar@[red]@{-}[d] & M \frac{1}{6} \ar@[red]@{-}[d] \\
M \frac{1}{3} \ar@[red]@{-}[r] & 3M \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & \boxed{6 M} \ar@[blue]@{-}[r] & 3M \frac{1}{2} \ar@[red]@{-}[d] \\ & M \frac{2}{3} & & M \frac{5}{6}} \]

To describe the moonshine group $(n|h)+e,f,\dots$ (an example was worked out in the tetrahedral snake post), we need to study the action of base-change with the matrix
\[
x = \begin{bmatrix} 1 & \frac{1}{h} \\ 0 & 1 \end{bmatrix} \]
which sends a lattice of the form $M \frac{g}{h}$ with $0 \leq g < h$ to $M \frac{g+M}{h}$, so is a rotation over $\frac{2 \pi M}{h}$ around $h M$. But, we also have to describe the base-change action with the matrix \[ y = \begin{bmatrix} 1 & 0 \\ n & 1 \end{bmatrix} \] and for this we better use the second description of the lattice as $M \frac{g}{h}=(\frac{g'}{h},\frac{1}{h^2M})$ with $g'$ the multiplicative inverse of $g$ modulo $h$. Under the action by $y$, the second factor $\frac{1}{h^2M}$ will be fixed, so this time we have to look at all lattices of the form $(\frac{g}{h},\frac{1}{h^2M})$ with $0 \leq g < h$, which again can be considered as another set of $h$-th roots of unity, centered at $hM$. Here's this second interpretation for $h=6$: \[ \xymatrix{M \frac{5}{6} \ar@[red]@{-}[d] & & 4M \frac{1}{3} \ar@[red]@{-}[d] & \\ 3M \frac{1}{2} \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & \boxed{6M} \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & 12 M \ar@[red]@{-}[r] \ar@[red]@{-}[d] & 4 M \frac{2}{3} \\ M \frac{1}{6} & 18 M \ar@[blue]@{-}[r] \ar@[blue]@{-}[d] & 36 M & \\ & 9M \frac{1}{2} & & } \] Under $x$ the first set of $h$-th roots of unity centered at $hM$ is permuted, whereas $y$ permutes the second set of $h$-th roots of unity.
These interpretations can be used to spot errors in computing the finite groups $\Gamma_0(n|h)/\Gamma_0(n.h)$.

Here’s part of the calculation of the action of $y$ on the $(360|1)$-snake (which consists of $60$-lattices).

First I got a group of order roughly $600.000$. After correcting some erroneous cycles, the order went down to 6912.

Finally I spotted that I mis-numbered two lattices in the description of $x$ and $y$, and the order went down to $48$ as it should, because I knew it had to be equal to $C_2 \times C_2 \times A_4$.

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