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Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics

Terry Gannon
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Monographs on Mathematical Physics
[Reviewed by
Fernando Q. Gouvêa
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Not too long ago, I got a call from someone in Colby’s admissions office. She told me that they had been updating their “lobby sheets,” descriptions of different departments that are provided to interested students, by adding the names and research interests of newly hired faculty. One of those new faculty members was a mathematician, and she had listed “moonshine” as one of her research interests. “Was this a joke?”, asked the person from admissions. I had to explain to her that yes, there really is a topic of mathematics research known as “moonshine,” and that no alcoholic beverages were involved.

It’s a strange name for a strange subject. It all started in the heady days of the discovery of some very strange finite simple groups, known as the sporadic groups. The biggest of these was nicknamed the “Monster.” At that point it was still not certain that the Monster group actually existed, but if it did, the dimensions of its (finitely many) irreducible representations were known. The smallest one, of course, is the trivial representation, of dimension d(0) = 1. Not very interesting. The next smallest had dimension d(1) = 196,883, and was already a fairly interesting object. (It later was the key to actually constructing the Monster, and therefore proving that such a group did exist.) The next two dimensions were d(2) = 21,296,876 and d(3) = 842,609,326. Your typical very boring very large numbers.

In another branch of mathematics entirely, there lived a modular function called j, usually written out as a power series in q = e2πiz:

j = q–1 + 744 + 196884q + 21493760q2 + 864299970q3 + … + c(n)qn + … 

The 744 is a normalization constant, not very important. The other coefficients, however, are canonical. John McKay was the first one to remark on how close d(1) = 196,883 is to c(1) = 196,884. Coincidence, or alien intervention?

Well, it’s even more remarkable:

c(1) = 196,884 = 196883 + 1 = d(1) + d(0)

c(2) = 21,493,760 = 21,296,876 + 196,883 + 1 = d(2) + d(1) + d(0)

c(3) = 864,299,970 = d(3) + d(2) + 2d(1) + 2d(0)

and so on. Each of the coefficients c(n) of the modular function j seemed to be a linear combination of positive small multiples of the dimensions d(n). This is what John H. Conway called “Monstrous Moonshine,” using the word in its British sense of  “Appearance without substance; something unsubstantial or unreal; … foolish or fanciful talk, ideas, ….” (OED, definition 2a)

John Thompson pointed out that one way that this could have come about was if there existed a sequence of representations Vn of the Monster, of dimension c(n) and decomposing into a sum of irreducible representations. Of course, one can build such a thing in a dumb way: just make each Vn equal to the sum of c(n) copies of the trivial representation. So the point was that the Vn should be fairly natural objects that would decompose into sums of a few irreducible representations. One could package all the Vn together to make a graded representation of the Monster.

Thompson pushed the guess a little more: the dimension of a representation is just the trace of the image of the identity element of the Monster. What if we took another element g, looked at its images in the Vn, took traces, and made a series like the one for j? Would one get a modular functions again?

That was easy to test numerically, and it turned out that Thompson’s procedure does produce modular functions. At that point it begins to become clear that something more than coincidence is going on here. But what?

The question has generated an immense amount of very difficult mathematics that mixes (as Gannon’s subtitle says) algebra (especially groups, representations, and infinite-dimensional Lie algebras), modular forms, and mathematical physics (especially in the form of conformal field theory). Almost all of the original conjectures have now been proved, but most of the mathematicians involved feel the mystery is still not dispelled, mostly because some of the proofs seem to work by miraculous strokes of luck that call for further probing and explanation.

Terry Gannon’s Moonshine Beyond the Monster is an introduction to this fascinating subject and to the mathematics that relates to it. Gannon has tried to do something unusual here. Instead of attempting a formal account of the story with full proofs (which would have required either very stiff prerequisites or multiple volumes), Gannon wants to explain to us “what is really going on.” His book is like a conversation at the blackboard, with ideas being explained in informal terms, proofs being sketched, and unknowns being explored. Given the complexity and breadth of this material, this is exactly the right approach.

This doesn’t mean the book is easy to read. There is a lot in here, and most readers will find at least some parts heavy going. Those who know about modular forms may not know that much about affine Lie algebras or conformal field theory, after all. Those who find a brief and breezy explanation of group representation theory inviting might well have trouble with a similarly brief account of modular forms.

For those curious about the subject and willing to do a little work, however, it’s hard to imagine a better way to learn about the field. Gannon has taken the risk of sharing with us how he thinks about this. He points out that most of us understand things best at a qualitative level, and writes accordingly. It works very well. The result is informal, inviting, and fascinating.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He understands the modular forms part of all this pretty well.

 Introduction: glimpses of the theory beneath Monstrous moonshine; 1. Classical algebra; 2. Modular stuff; 3. Gold and brass: Affine algebras and generalisations; 4. Conformal field theory: The physics of Moonshine; 5. Vertex operator algebras; 6. Modular group representations throughout the realm; 7. Monstrous Moonshine; Epilogue; References.