Pour Me Some of the Good Stuff: A brief introduction to Moonshine

Mathematics is an absolutely enormous field, spanning thousands of years and millions of pages of “fundamental” reading for anyone wishing to master all of its major aspects. As such, it is convenient to divide math into several sub-fields (algebra, geometry, number theory, analysis, etc.). Most mathematicians, even the best ones, only work in one or two of them for their entire career.

These sub-fields often seem completely unrelated to one another, with their only commonality being that they all adhere to the same basic axioms. But sometimes, when the weather is right, and the planets align, a discovery is made that reminds everyone just how interconnected the complex web of mathematics truly is. Perhaps the most astonishing of these connections is Moonshine: a set of observations, conjectures, and theorems that connect functions arising in number theory with certain structures in group theory. What’s perhaps even more interesting is that the structures involved in explaining these connections are intimately related to 21st century physics, in particular, string theory.

Historically, the first known and most well understood example of a Moonshine involves the modular $j$ function and the Monster group. The former is a very well-studied function arising in number-theoretic contexts, while the latter is a massively complicated algebraic structure and the largest of the so-called sporadic finite simple groups. The $\displaystyle j$ function can be written as a Fourier series in a complex variable $\displaystyle \tau$ , and the first few coefficients are given by

$\displaystyle j(\tau)=q^{-1}+744+196884q+\cdots,$

where $\displaystyle q=e^{2\pi i \tau}$ . On the algebraic side, the Monster group, just as any group, can be represented by a large set of matrices, with group composition being represented by matrix multiplication. The smallest dimension of these matrices such that one can faithfully reconstruct the Monster group from its representation is $196883$ . The attentive reader might notice that this number is shockingly close to the order $q$ coefficient of the $j$ function. Indeed, the first so-called McKay equation is simply the observation that

$196884=196883+1.$

The McKay equation, while incredibly simple to prove, begs explanation. Why should a function studied heavily in number theory have anything to do with an exceptionally large algebraic structure? The two, at first glance seem as though they should be as far separated as two objects in mathematics can get. We could, of course, dismiss the McKay equation as a coincidence, but it turns out that there is an infinite list of equations similar to the one above that relate the higher-order coefficience of the $j$ function to higher-dimensional representations of the Monster group. These relations lie at the heart of the special case of Moonshine, known as Monstrous Moonshine. In this post, we will explore some of the basic ideas at play here, as well as a brief discussion of the modern understanding of Monstrous Moonshine in the context of strinig theory.

The Modular j Function

The first half of the Monstrous Moonshine puzzle is the very famous (among mathematicians) modular $j$ function (also known as the Klein $j$ invariant). A full explanation of this special function would require a bit more than this blog post has to offer, but the basic ideas are rather simple.

The $j$ function is a function that naturally lives on the upper-half plane — the subset of complex numbers whose imaginary part is positive. That is,

$\mathbb{H}=\{a+bi|b>0\},$

where $i=\sqrt{-1}$ is the imaginary unit. The $j$ function takes in a variable $\tau$ and outputs some complex number.

There is a superbly large set of functions that live on the upper-half plane. What makes the $j$ function special is that it is a modular invariant, meaning that it behaves particularly well under a certain set of transformations known as modular transformations. A modular transformation is a map from the upper-half plane to itself that acts as

$\tau\to\displaystyle\frac{a\tau+b}{c\tau+d},$

where $a$ , $b$ , $c$ , and $d$ are integers satisfying $ad-bc=1$ . We typically write modular transformations as $2\times 2$ matrices, and write their action as

$\displaystyle\begin{pmatrix}a & b \\ c & d\end{pmatrix}\cdot\tau\equiv\frac{a\tau+b}{c\tau+d}.$

The benefit of writing modular transformations as matrices is that applying two modular transformations in a row is compatible with matrix multiplication. The resulting group of transformations is given the fancy name $\text{SL}_2(\mathbb{Z})$ .

When we say that the $j$ function is modular, we mean that $j$ behaves nicely under modular transformations. Specifically, we want

$j\left(\gamma\cdot\tau\right)=j(\tau)$

for any matrix $\gamma\in\text{SL}_2(\mathbb{Z})$ . This alone doesn’t uniquely determine $j$ , but requiring that $j$ behaves nicely as the imaginary part of $\tau$ gets very large (as well as a few other requirements) uniquely determines $j$ up to normalization.

The modularity of $j$ also implies that $j$ is periodic. In particular, we have

$j(\tau+1)=j(\tau).$

Thus, $j$ admits a Fourier series description. In particular, if we let $q\equiv e^{2\pi i\tau}$ , then we can write $j$ as a Laurent series in $q$ (this is the definition of a Fourier series with period 1). Explicitly calculating this function, setting the constant term to $744$ (a convenient choice), and normalizing the $q^{-1}$ term to have a unit coefficient, the Fourier series of $j$ can be found to be

$j(\tau)=q^{-1}+744+196884 q+\cdots.$

As it turns out, all functions that behave “nicely” under modular transformations can be written as the ratio of polynomials in the $j$ function, and such functions are central to the study of elliptic curves, which has countless applications to number theory (as an example, such concepts play a central role in the proof of Fermat’s Last Theorem).

As an aside, the modular $j$ function has another neat quality that isn’t directly related to Moonshine. If a positive number $n$ satisfies certain properties (namely that the field extension $\mathbb{Q}[\sqrt{-n}]$ is a unique factorization domain), then

$\displaystyle j\left(\frac{1+\sqrt{-n}}{2}\right).$

is an integer. Since $j((1+\sqrt{-n})/2)\approx e^{\pi\sqrt{n}}$ for large $n$ (just plug the argument into the Fourier series), then for these numbers, $e^{\pi\sqrt{n}}$ is nearly an integer. Particularly, for the largest value of $n$ satisfying these conditions, namely $n=163$ , we have the near-integer

$e^{\pi\sqrt{163}}\approx 640320^3+744,$

which holds to surprisingly high precision.

The Scary Thing in the Closet

The second half of the story of Monstrous Moonshine is the titular entity — the Monster group. To begin, a group is a set of elements $G$ with a specified rule (we’ll denote by multiplication $a\cdot b$ ) for how to take two elements and make a third. This rule has to satisfy a few fundamental properties:

The product of two elements in $G$ needs to be another element in $G$ .
There must be an element called $1$ such that $a\cdot 1=1\cdot a=a$ for all $a$ in $G$ . This is known as the identity element.
Any three elements must satisfy $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ .
For any element $a$ , there must exist an inverse element $a^{-1}$ such that $a\cdot a^{-1}=a^{-1}\cdot a=1$ .

A lot of these properties should remind the reader of regular numbers. Indeed, if we consider the real numbers without zero (denoted as $\mathbb{R}\setminus\{0\}$ ) obey all of these axioms under standard multiplication. However, the main difference between a generic group and the real numbers is that one is not necessarily guaranteed that $a\cdot b=b\cdot a$ . For example, the set of non-singular $n\times n$ matrices form a group under matrix multiplication, for which the order of multiplication matters.

Groups which have a finite number of elements are called finite groups. From the beginning of the 20 $^{\text{th}}$ century, the classification of all finite groups was a large unsolved problem. This classification is made much easier by considering simpler finite groups, from which other, more complicated groups can be formed by a kind of multiplication of groups. As such, these groups are called simple groups, and play the role of prime numbers in the study of groups (in the sense that other groups can be built up from them). Thus, one of the biggest challenges of 20 $^{\text{th}}$ century mathematics was the classification of finite simple groups. This classification has since been completed, and its proof spans several decades, dozens of authors, and thousands of pages. The basic result is that every finite simple group either belongs to

One of several “regular” infinite families.
One of 26 groups that do not fit into any of the above families, known as the sporadic groups.

The groups that are of interest in Moonshine are the sporadic groups, which are, in a sense, the misfits of group theory. Of these 26, the largest is of particular importance. It has around $8\times 10^{53}$ elements, and is given the well-deserved title of the Monster group $\mathbb{M}$ .

Perhaps the most interesting thing about the monster group is just how elusive it is. Many of the groups we know can be interpreted as describing the symmetries of some geometric object, or an interesting set of matrices acting on a vector space. For instance, the symmetries of a regular $n$ -gon form a group called the dihedral group $D_n$ , and such a group has a clear geometric interpretation. A natural question is what, if any, such interpretation the Monster group has.

One such “representation” of the Monster group involves writing each element of the Monster group as a large matrix. The smallest such dimension for which this is possible when the matrices have real coefficients is $196884$ . In mathematical jargon, we say that the smallest faithful real-valued representation of $\mathbb{M}$ has dimension $196884$ . This, of course, falls short of an intuitive explanation of what exactly the Monster group is, but the study of these representations reveals a great deal about the structure of the Monster (or any other group, for that matter).

A Stringy Explanation

Now that we’ve seen a bit more about each of the two sides of the Moonshine puzzle, we are still left with a need for an explanation. In fact, the more one knows about each side of the Moonshine equation, the less sense the connection seems to make. Why should the dimensions of the representations of the Monster group be related to the Fourier coefficients of the $j$ function?

As it turns out, within the context of modern physics, there is a somewhat natural way to relate these two vastly different beasts which arises in the study of a very specific scenario in string theory.

String theory is a modern mathematical model for a physical theory of the universe that can (possibly) unify the three forces of nature that are compatible with quantum theory (electromagnetism and the two nuclear forces) with gravity. The basic idea of string theory is to replace the concept of the point-like particles that make up our current understanding of the world with one-dimensional string-like fundamental objects. Such strings are allowed to vibrate and move around, and their motion and vibration are proposed to make up the particles we see in our current experiments. That is, according to string theory, if you were able to zoom in far enough, all particles would look like tiny vibrating strings.

As people (mostly Witten) starting pointing out in the 1980s, string theory is a natural home for many ideas in modern mathematics, including algebraic topology, algebraic geometry, and differential geometry, just to name a few. In fact, as it turns out, once we pass to the quantum version of string theory (just as we pass to the quantum version of particle theory to obtain the Standard Model), the theory becomes a machine for churning out modular-invariant functions (such as the $j$ function). This is because when we compute the most basic quantity in string theory (the so-called partition function), one of the contributions ends up being an integral over all possible torus shapes. The shape of a torus can be described by a complex number $\tau$ that lives in the upper-half plane, and two tori are equivalent if their value of $\tau$ (the so-called modulus) differ only by a modular transformation. Because computing the partition function in string theory involves integrating over all inequivalent $\tau$ , the integrand, as well as the integration measure need to be modular invariants. (That was a lot to absorb without context from string theory, but I’ll probably write another post on it in the future. For now, just know that string theory acts as a machine for computing modular invariants.)

String theory as the origin of Moonshine.

Now, specifying a string theory (at least a bosonic one — the simplest case) involves specifying the spacetime in which it moves. It’s possible to consider string theory moving on regular flat spacetime, but the most interesting scenarios happen when we allow the string to move on a nontrivial geometry. For instance, we can consider a string moving on an $n$ -dimensional torus, which can be described as $\mathbb{R}^n/L$ for some lattice $L$ .

A fantastic thing happens when we pick a very specific lattice $L$ , known as the Leech lattice, which is the lattice that gives the optimal packing of spheres in 24 dimensions. The Leech lattice is interesting here because one can actually think of the Monster group as a certain set of symmetries of the Leech lattice. Now, if we consider string theory on $\mathbb{R}^{24}/L$ (plus a bit more structure), where $L$ is the Leech lattice, then the partition function on a torus of modulus $\tau$ is given almost exactly by the $j$ function! This is essentially how Monstrous Moonshine arises in string theory: the Monster group represents a certain set of symmetries of the spacetime in which the string moves, and the $j$ function is the modular invariant that this theory churns out. With a bit of math, one can use this fact to demonstrate the relationship between the dimensions of representations of the Monster group and the Fourier coefficients of the $j$ function!

Moonshine Beyond the Monster

Monstrous moonshine was the first noted example of a moonshine. However, there are many more examples that are currently being explored by mathematicians and physicists alike. Most of them focus on the other 25 sporadic groups that are smaller than the monster group, and their connections to modified versions of modular forms. Though few of the new moonshine conjectures have been proven, it still seems that the mathematics of string theory will serve as a source of intuition and guidance in the work still to be done.

For anyone curious about the details of Moonshine and how it relates to string theory, go no further than Terry Gannon’s wonderful book “Moonshine Beyond the Monster,” from which I shamelessly borrowed all of the things I currently know about moonshine.

Pour Me Some of the Good Stuff: A brief introduction to Moonshine

The Modular j Function

The Scary Thing in the Closet

A Stringy Explanation

Moonshine Beyond the Monster

Published by Bob Knighton

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The Modular j Function

The Scary Thing in the Closet

A Stringy Explanation

Moonshine Beyond the Monster

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Published by Bob Knighton

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