number theory

Moonshine Link Discovered for Pariah Symmetries

A type of symmetry so unusual that it was called a “pariah” turns out to have deep connections to number theory.

Peter Diamond for Quanta Magazine

In 1892, the mathematician Otto Hölder posed a question that would occupy the field for more than a century: Is it possible to make a periodic table of all finite symmetry? The answer, to which hundreds of mathematicians have contributed, is yes. But the taxonomy that emerged from this monumental effort has prompted both enlightenment and head scratching. For in addition to the well-understood elements of the symmetry chart, a handful of outliers made themselves known — elements mathematicians could prove must exist but couldn’t connect to any natural shapes.

In particular, mathematicians discovered six maverick forms of symmetry that sit so far out on the fringe of the symmetry world that they became known as “pariahs.” When the first pariahs were discovered in the mid-1960s, wrote the mathematician Felipe Zaldivar in 2010, “one could almost hear the ‘Who ordered that?’”

Apart from their cameo role in the classification of finite symmetries, the pariahs “have not appeared anywhere in mathematics,” wrote Ken Ono, a mathematician at Emory University, in an email. “They are something like the super heavy metals in the periodic table of elements.”

Now Ono, John Duncan of Emory, and Michael Mertens of the University of Cologne in Germany have succeeded in welcoming one of the pariahs, called the O’Nan group, into the framework of a theory known as “moonshine.” Originally developed decades ago for a gargantuan symmetry structure called the monster group, moonshine forges deep connections between groups of symmetries, models of string theory and objects from number theory called modular forms.

O’Nan moonshine is “part of the wave of sort of a new paradigm of moonshine research that has been bubbling up in the past year,” said Miranda Cheng, a mathematician and physicist at the University of Amsterdam and France’s National Center for Scientific Research. “Every week we’re discovering something new.”

The new work, which the researchers describe today in Nature Communications, puts the O’Nan group at the crest of this new wave of moonshine, which links certain symmetry groups to special classes of “weight 3/2” modular forms, objects that also show up in natural counting functions for black holes and for higher-dimensional generalizations of strings called “branes.” Because of these and other confluences, wrote the Stanford University physicist Shamit Kachru in an email, “my guess is the subject will have legs and develop well.”

There’s every reason to hope, Duncan said, that this new wave of moonshine may soon illuminate the dark corners of mathematics where the other pariahs lie hidden.

And the particular modular forms that appear in O’Nan moonshine connect it to some of the most central objects in number theory, including elliptic curves, which played a starring role in Andrew Wiles’ 1994 proof of Fermat’s Last Theorem. “The O’Nan group has unexpectedly been found to hover over some of the questions which have been of interest to number theorists for a super long time,” Ono wrote.

The Moonshine Landscape

Mathematicians generally think about a shape’s symmetry in terms of the group of transformations that leave its geometry intact — so an equilateral triangle has a symmetry group consisting of three rotations and three reflections, while a circle has a symmetry group consisting of all possible rotations or reflections through its center. Finite symmetry groups, like that of the triangle, can always be built up out of what are called “simple” groups — atoms of symmetry that can’t be divided into smaller groups.

For example, the equilateral triangle’s six-member symmetry group is not simple, but it can be constructed by combining two simple groups: a three-member group consisting of the triangle’s rotations, and a two-member group that specifies whether to reflect the triangle after rotating or leave it alone. Every finite group has a unique molecular formula of this kind — a collection of simple groups from which it is made.

Many of the simple groups are well-understood and easy to describe. But when mathematicians tried to make a periodic table of all finite simple groups, they were electrified to discover 26 “sporadic” groups that fit none of the familiar molds. Twenty of these groups cluster into what’s called the “happy family,” headed by the monster. And six groups are left out in the cold — the pariahs.

“It would have been comforting, poetic, neat, etc., to say that every sporadic group is part of a single ‘master object’ which unifies and explains the whole business with a single context, but it did not work out that way,” wrote Robert Griess in an email. Griess, a mathematician at the University of Michigan, constructed the monster group in 1982 and gave the happy family and the pariahs their names. (He also named the monster the “friendly giant,” but for some reason that name didn’t stick.)

By sheer accident, mathematicians quickly found a home for the monster group in the broader mathematical universe. In 1978, John McKay of Concordia University in Montreal noticed that the same number — 196,884 — occurs in two widely different mathematical contexts. One is as a combination of two numbers from the monster group, and the other is as a coefficient of the “j-function,” one of the simplest examples of a modular form — a type of function with repeating patterns like those in Escher’s circular angels-and-devils tilings.

The idea that these two far-flung areas of mathematics could be connected seemed so fantastic that it became known as moonshine. But more numerical “coincidences” started piling up, and eventually mathematicians figured out a deep reason for them: The monster group and the j-function are connected via string theory. In a particular 24-dimensional string theory world, the j-function’s coefficients capture how strings can oscillate, while the monster controls the underlying symmetry.

This “monstrous” moonshine showed that the monster group isn’t just some anomalous object forced into existence by abstract considerations. It is the symmetry group of a natural space, and it is closely connected to modular forms, which number theorists have been studying for centuries. The development gave rise to entirely new areas of mathematics and physics, and it earned Richard Borcherds, of the University of California, Berkeley, a Fields Medal in 1998.

For decades, monstrous moonshine seemed like a one-off phenomenon. But in 2010, physicists started noticing that if they looked at groups related to certain 24-dimensional lattices, a raft of new numerical coincidences emerged. By 2013, Cheng, Duncan and the physicist Jeffrey Harvey of the University of Chicago had conjectured the existence of 23 more moonshines — one for each lattice — that included several more members of the happy family as well as other symmetry groups. Two years later, Duncan, Ono and the mathematician Michael Griffin proved that these moonshines do exist. But while moonshine gradually spread its beams further over the happy family, the pariahs remained in the shadows.

With this new list of 23 moonshines, researchers thought at first that they had fully fleshed out the possibilities, Harvey said. “We had a nice tidy classification that seemed complete because of the link to these lattices,” he wrote in an email. But these weren’t, in fact, the only moonshines out there. “In the middle of research when everything is a jumble and confusing it can be hard to step back and realize you are missing a nice general idea,” he wrote.

Now, mathematicians and physicists have embarked on what has the makings of a third wave of moonshine, centered around weight 3/2 modular forms. “Weighted” modular forms, instead of repeating exactly on each angel and devil in an Escher tiling, get multiplied by a particular factor as you go from one angel or devil to the next. Besides the new O’Nan moonshine, Cheng said, several more papers are in the works that link weight 3/2 modular forms to a wide variety of symmetry groups, including M11 and M23, two of the monster’s descendants in the happy family.

The new moonshine “really smells like a different game,” Cheng said. “The connection between finite groups and modular forms is really general, much more general than we thought.”

The rapid pace of recent discoveries is generating a mix of excitement, confusion and frustration, Harvey said — frustration, because researchers have not yet found the string theory models that would make sense of these new correspondences between symmetry groups and modular forms. “I think there’s some mysterious class of objects that will explain a lot of this, but we don’t know what they are yet,” he said. “We just have hints that they exist.”

Many other possible weights have yet to be examined, said Harvey, who together with Brandon Rayhaun of Stanford recently discovered a weight 1/2 moonshine for the Thompson group, one of the monster’s grandchildren.

“Where does it stop? The answer, as far as I know, is that nobody knows,” he said. “People are sort of exploring, trying to understand how big the landscape is.”

Pariah Moonshine

Duncan came upon the beginnings of O’Nan moonshine in much the same way that McKay had stumbled upon the original monstrous moonshine almost 40 years earlier. Duncan noticed that a certain dimension in which the O’Nan group has a special representation — 26,752 — is the same as the first important coefficient of a weight 3/2 modular form he’d bumped into in earlier work on Thompson moonshine. Weight 3/2 modular forms hadn’t been the main characters in most previous moonshines, but Duncan and Mertens soon became convinced that this particular modular form was the key to creating moonshine for the O’Nan group (which is named for its discoverer, Michael O’Nan, who passed away on July 31).

Duncan happened to describe the new moonshine to Ono one evening, over dinner with their families. Ono had never heard of the O’Nan group, but he immediately recognized the modular forms involved. “These forms are like old friends to me,” he wrote by email.

Ono realized that they belonged to a special collection of weight 3/2 modular forms that the number theorists Benedict Gross, Winfried Kohnen and Don Zagier had shown in 1987 to be intimately related to many of the central problems in number theory, such as counting special points on certain elliptic curves. Some of the other new moonshines under development tie into number theory in a similar way, Cheng said.

The connection Gross, Kohnen and Zagier uncovered involves “some of the hottest stuff” that number theorists have been studying in the past few decades, Ono said. “None of us had any idea that there would be … these strange finite groups lurking in the background.”

The new moonshine means that the O’Nan group captures something about the symmetries of these number-theoretic objects — but just what it captures may not become fully clear until the right string theory model for O’Nan moonshine is found, Duncan said. “If there is a good physical answer to that question, then that’s potentially bringing physical techniques in to bear on these deep number-theoretic problems,” he said. “I think that’s a really exciting area to try and make progress on next.”

What’s already evident, though, is that the O’Nan group isn’t as much of a pariah as mathematicians thought. And even though researchers are just starting to explore the broader landscape of moonshine, some are already asking themselves whether, down the road, these new moonshines might finally offer a satisfying explanation for just what the sporadic groups — the monster, its kin and the pariahs — are doing in the taxonomy of finite symmetry in the first place. Perhaps, after all, the sporadic groups will turn out to be examples of some natural phenomenon, Duncan said, if we look at them through the right lens.

“There’s starting to be a possibility that moonshine will really encompass all of the sporadic groups,” Harvey said, “and perhaps sort of explain why these oddballs are there.”

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