Pariahs are fundamental building blocks in a branch of mathematics called group theory, but seem to be unconnected from both physics and other areas of mathematics. Such a connection has now been identified.

The idea of a group is intrinsic to mathematics — it is simply a collection of actions called elements. For example, the symmetries of an equilateral triangle form a group consisting of six elements (three reflections and three rotations), and the shuffling of a deck of 52 playing cards forms a group that has about 8 × 1067 elements (the different ways in which the cards can be arranged). If something is fundamental to mathematics, then it is usually fundamental to physics. Indeed, the Lorentz group is central to Einstein's special theory of relativity, and the gauge group is central to the standard model of particle physics1. However, certain groups called pariahs were thought to have no connection to the physical world. Writing in Nature Communications, Duncan et al.2 report the discovery of such a connection, which could have implications for both mathematics and physics.

Points on a plane are identified using their x, y coordinates. Because these coordinates are a pair of numbers, a plane can be referred to as 2-space. Similarly, we can speak of 3-space (if we include a third dimension), 4-space (if we also include time), and so on. Groups can act on n-space (where n is any number between 1 and infinity) by, for example, rescaling, rotating or reflecting points. These actions, known as representations, are well understood and computer-friendly, and feature in many areas of mathematics and physics. For example, every particle in high-energy physics corresponds to a representation of the Lorentz group1.

Humans think reductively: we understand something complicated in terms of its basic components. Like the clicking together of Lego blocks, a large group can be obtained by clicking together smaller (usually simpler) groups. We do this by putting the smaller groups side-by-side, and then allowing one-way communication between them — analogous to fitting the prongs of one Lego block into the underside of another. The archetypal example is the addition of multi-digit numbers: when adding together 27 and 45, we first add 7 and 5 in one column to get 12, 'carry' the 1 and then add 1, 2 and 4 in a second column. In doing so, we click together two copies of what is known as the addition modulo 10 group (one copy for each column), with one-way communication taking place through the 'carry' process.

Just as we can write any number as a product of prime numbers (for example, 60 = 22 × 3 × 5), we can write any group as a clicking together of so-called simple groups. To some extent, group theory can be reduced to understanding the simple groups (the Lego blocks) and the different ways that they can be clicked together. One of the great accomplishments of twentieth-century mathematics was the determination of the complete list of simple groups that contain a finite number of elements3. Almost all of these groups belong to one of 18 'infinite families' — for example, the nth simple group in one of the families consists of half of the ways in which n playing cards can be arranged. But there are also 26 isolated groups called the sporadics.

Duncan and colleagues' work concerns these sporadics. The largest is known as the monster; it contains about 8 × 1053 elements3 and all but 6 of the other sporadics. The remaining 6 are jokingly called the pariahs. It could have been that every sporadic is completely disconnected from all other areas of mathematics and science. But the monster is not irrelevant: its representations show up in modular functions4.

Modular functions are to complex numbers what periodic functions, such as sine and cosine, are to real numbers. (Complex numbers are quantities expressed in the form a + bi, where a and b are real numbers and i is the positive square root of −1.) More precisely, curves based on complex numbers can be visualized as surfaces, and functions on most complex curves are modular functions. The observation that certain modular functions are built up from representations of the monster was dubbed monstrous moonshine.

This discovery was completely unexpected. Nowadays, mathematicians explain the existence of monstrous moonshine by proposing that there is a two-dimensional quantum field theory, related to string theory, whose symmetry is the monster. Modular functions arise naturally in string theory, partly because the strings trace out surfaces as they move. In 1998, the mathematician Richard Borcherds was awarded the Fields Medal, the highest honour in mathematics, for his contribution to this work5 (the story is described in ref. 6).

Other moonshines, relating other sporadic groups to functions closely related to modular functions, have been discovered over the years. For example, Mathieu moonshine7 involves a sporadic group that is called, by some, the most remarkable group of all, because it features in many different contexts. These moonshines all seem related to string theory, but the specific connections remain mysterious. Crucially, none have involved the pariahs. Mathematicians therefore began to suspect that perhaps the pariahs really were outcasts — Lego blocks too strange to be relevant.

Building on their earlier work8, Duncan et al. discovered a connection between the representations of a pariah called the O'Nan group9 and the theory of elliptic curves — complex curves shaped like the surfaces of doughnuts. The O'Nan group has approximately 5 × 1011 elements9, which is about average for a pariah. Its representations give modular forms (closely related to modular functions) that contain intricate information about elliptic curves.

“The authors' moonshine has a rather different flavour from that of the earlier discoveries.”

Duncan and colleagues' moonshine has a rather different flavour from that of the earlier discoveries. In particular, it is more difficult to see a possible connection to string theory. Because of this, the role that the O'Nan group has in nature remains unclear. Moreover, the precise relationship of its representations to elliptic curves needs to be fleshed out. The authors' discovery also raises the question of whether moonshines exist for the other pariahs, but this will require further work.

It is always difficult to gauge the importance of a mathematical result without the hindsight that many years brings. Nevertheless, Duncan et al. have shown us a door. Whether it is to a new closet, house or world, we cannot yet say, but the results are certainly unexpected, and no one will think of the pariahs in the same way again.

References 1 Weinberg, S. The Quantum Theory of Fields Vol. 1 (Cambridge Univ. Press, 2005). 2 Duncan, J. F. R., Mertens, M. H. & Ono, K. Nature Commun. 8, 670 (2017). 3 Elwes, R. Plus Mag.; https://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups (2006). 4 Conway, J. H. & Norton, S. P. Bull. Lond. Math. Soc. 11, 308–339 (1979). 5 Borcherds, R. E. Invent. Math. 109, 405–444 (1992). 6 Gannon, T. Moonshine Beyond the Monster (Cambridge Univ. Press, 2006). 7 Eguchi, T., Ooguri, H. & Tachikawa, Y. Exp. Math. 20, 91–96 (2011). 8 Duncan, J. F. R., Mertens, M. H. & Ono, K. Preprint at https://arxiv.org/abs/1702.03516 (2017). 9 O'Nan, M. E. Proc. Lond. Math. Soc. 3rd Ser. 32, 421–479 (1976). Download references

Author information Affiliations the Department of Mathematical Sciences, University of Alberta, Edmonton, T6G 2G1, Alberta, Canada Terry Gannon Authors Terry Gannon View author publications You can also search for this author in PubMed Google Scholar Corresponding author Correspondence to Terry Gannon.

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