Curiosities: Pursuing the Monster

What lies beneath a structure with an unimaginable 196,883 dimensions?

In 1981, Freeman Dyson addressed a typically distinguished group of scholars gathered at the Institute for a colloquium, but speaking on a decidedly atypical subject: “Unfashionable Pursuits.”

The problems which we face as guardians of scientific progress are how to recognize the fruitful unfashionable idea, and how to support it. To begin with, we may look around at the world of mathematics and see whether we can identify unfashionable ideas which might later emerge as essential building blocks for the physics of the twenty-first century.*

He surveyed the history of science, alighting eventually upon the monster group—an exquisitely symmetrical entity within the realm of group theory, the mathematical study of symmetry. For much of the twentieth century, mathematicians worked to classify “finite simple groups”—the equivalent of elementary particles, the building blocks of all groups. The classification project ultimately collected all of the finite simple groups into eighteen families and twenty-six exceptional outliers. The monster was the last and largest of these exceptional or “sporadic” groups.

The first of the sporadic simple groups was discovered in the nineteenth century by French mathematician Émile Mathieu. It wasn’t until 1973 that two mathematicians—Bob Griess at the University of Michigan and Bernd Fischer at Universität Bielefeld—independently predicted the existence of the monster. They did this in a manner similar to how physicists predicted the existence of the Higgs boson, the quantum of the Higgs field molasses that pervades the ether and endows elementary particles with mass. And just as physicists long hunted the Higgs boson, so too did the prediction of the monster send mathematicians hunting for information, confirmation, any crumbs or clues about the monster’s existence.

It didn’t take long before John Conway, then at Cambridge, now at Princeton University, came back with the monster’s order, its number of symmetries:

8 · 1053
246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

And possessing these 808 sexdecillion or so symmetries—not infinity by any means, but heading in that direction—the monster certainly did not reside in anything close to our three-dimensional space. The monster lives, or more precisely acts, in an unimaginable 196,883 dimensions.

Further clues came in 1978 when Concordia University’s John McKay noticed that 196,883+1=196,884—elementary addition, perhaps, but 196,884 was a number of considerable significance in modular functions, a faraway land on the other side of the mathematical ocean. Conway and Simon Norton (then both at Cambridge) pursued McKay’s observation as more than mere numerology. They proposed their monstrous moonshine conjectures—“moonshine” because the conjectures seemed illicit and illegal, as well as illuminating—and marshaled evidence to support the unexpected relationship between these two mathematical structures. The moonshine conjectures also postulated that given the evidence, there should be something underpinning the monster. Conway had discovered a group, Co1, in 24-dimensional space that was underpinned by the Leech lattice, a structure that arises in number theory and coding theory. Beneath the Mathieu group M24 lay the error-correcting Golay code. What underlay the monster?

In 1981, a few days before his talk, Dyson received in the mail the final installment of a long paper by Griess confirming the monster’s existence. Griess, while a Member at the Institute (1979–80, 1981, 1994), had constructed the monster as a group of rotations in 196,883-dimensional space (and in the process producing the Griess algebra expressly for that purpose). Conway later simplified this construction​—one among many re-imaginings of the monster to follow.

“What has all this to do with physics?” Dyson asked in his talk.

Probably nothing. Probably the sporadic groups are merely a pleasant backwater in the history of mathematics, an odd little episode far from the mainstream of progress. We have never seen the slightest hint that the symmetries of the physical universe are in any way connected with the symmetries of the sporadic groups. So far as we know, the physical universe would look and function just as it does whether or not the sporadic groups existed. But we should not be too sure that there is no connection. Absence of evidence is not the same thing as evidence of absence. Stranger things have happened in the history of physics than the unexpected appearance of sporadic groups. We should always be prepared for surprises. I have to confess to you that I have a sneaking hope, a hope unsupported by any facts or any evidence, that sometime in the twenty-first century physicists will stumble upon the monster group, built in some unsuspected way into the structure of the universe. This is of course only a wild speculation, almost certainly wrong.

But Dyson, as it turns out, is almost certainly right.

At the International Congress of Mathematicians in the summer of 1998, Richard Borcherds, now at Berkeley (previously at Cambridge and previously a Ph.D. student of Conway’s), received the Fields Medal for his proof of the ­moonshine conjectures. At the ceremony, mathematical physicist Peter Goddard—Director (2004–12) and now a Professor at the Institute—delivered the laudation. Borcherds in his proof had made critical use the “no-ghost theorem” by Goddard and Charles Thorn (at the University of Florida). “Displaying penetrating insight, formidable technique, and brilliant originality, Richard Borcherds has used the beautiful properties of some exceptional structures to motivate new algebraic theories of great power with profound connections with other areas of mathematics and physics,” said Goddard. “He has used them to establish outstanding conjectures and to find new deep results in classical areas of mathematics. This is surely just the beginning of what we have to learn from what he has created.”

Borcherds’s creation fulfilled Dyson’s hope that the monster would somehow be embedded into the structure of the universe, or at least took it a step in that direction. His proof demonstrated that the monster is the symmetry group not of a lattice or a code but of conformal field theory, part of the mathematical language of string theory.

For some, this is raison d’etre enough for the monster. For others, like Conway, the monster remains a mystery. Conway has tried to read some of the work linking the monster to conformal field theory, but he doesn’t find it helps with the question of why the monster exists. In his view, conformal field theory is too complicated to understand, and thus too complicated to be the only answer.

Dyson, for his part, offered another possibility in concluding his talk:

The only argument I can produce in its favor is a theological one. We have strong evidence that the creator of the universe loves symmetry, and if he loves symmetry, what lovelier symmetry could he find than the symmetry of the monster? 

* The colloquium was sponsored by the Humboldt Foundation, and Dyson’s talk was reprinted in its entirety in The Mathematical Intelligencer, Vol. 5, No. 3, 1983.

Recommended Reading: Read "Mathematicians Chase Moonshine's Shadow" by Erica Klarreich (March 12, 2015) in Quanta Magazine:

Siobhan Roberts, a Director’s Visitor at the Institute, is currently finishing a biography of Princeton mathematician John Horton Conway, to be published by Bloomsbury in spring 2015. She is the author of King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry, which won the Mathematical Association of America’s 2009 Euler Prize for expanding the public’s view of mathematics.