In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular the j function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979. The term "moonshine" was used by Conway in the sense of "foolish or crazy ideas", reflecting the seemingly absurd nature of the connection; as number theorist Don Zagier later put it, "they called it moonshine because it appeared so far-fetched."
The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.
History
The Jack Daniel's Problem
In the mid-1970s, mathematicians Jean-Pierre Serre, Andrew Ogg and John G. Thompson studied the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of the Hecke congruence subgroup Γ0(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has genus zero exactly for p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg attended a lecture by Jacques Tits in which the conjectural order of the monster group was presented, he noticed that these were precisely the prime factors of the group's size. He published a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this coincidence. This observation is widely considered the earliest hint of monstrous moonshine, and the challenge of explaining it became known as "The Jack Daniel's Problem". These 15 primes are now known as the supersingular primes, not to be confused with the use of the same phrase with a different meaning in algebraic number theory.
Although Richard Borcherds's 1992 proof of the monstrous moonshine conjecture established a deep connection between the monster group and modular functions, it provides a route from the monster to the genus-zero property but not in the reverse direction, leaving Ogg's original question not fully resolved. In 2014, John Duncan and Ken Ono revisited the problem, showing that the moonshine functions for order p elements of the monster yield the set of characteristic p supersingular j-invariants (apart from 0 and 1728), and discussing the coincidence using the first principles of moonshine. , the bottle of Jack Daniel's remains unclaimed.
McKay's observation and the Conway–Norton conjecture
In 1978, John McKay found that the first few terms in the Fourier expansion of the normalized J-invariant could be expressed in terms of linear combinations of the dimensions of the irreducible representations of the monster group M with small non-negative coefficients. Specifically, McKay noticed that the coefficient 196884 was precisely one more than 196883, the degree of the smallest faithful complex representation of the monster group. The J-invariant is
with and τ as the half-period ratio, and the M expressions, letting = 1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, ..., are
The left-hand side of the equations is the coefficients of , while in the right-hand side the integers are the dimensions of irreducible representations of the monster group M. (Since there can be several linear relations between the , such as , the representation may be in more than one way.)
McKay viewed this as evidence that there is a naturally occurring infinite-dimensional graded representation of M, whose graded dimension is given by the coefficients of J, and whose lower-weight pieces decompose into irreducible representations as above. After he informed John G. Thompson of this observation, Thompson suggested that because the graded dimension is just the graded trace of the identity element, the graded traces of nontrivial elements g of M on such a representation may be interesting as well.
When told by McKay that the coefficient of (namely 196884) was precisely one more than the degree of the smallest faithful complex representation of the monster group (namely 196883), Conway reportedly replied that this was "moonshine" (in the sense of being a crazy or foolish idea). The term "monstrous moonshine" thus refers both to the monster group M and to the perceived lunacy of the intricate relationship between M and the theory of modular functions.
Conway and Norton computed the lower-order terms of such graded traces, now known as McKay–Thompson series Tg, and found that all of them appeared to be the expansions of Hauptmoduln. In other words, if Gg is the subgroup of SL2(R) which fixes Tg, then the quotient of the upper half of the complex plane by Gg is a sphere with a finite number of points removed, and Tg generates the field of meromorphic functions on this sphere.
Based on their computations, Conway and Norton produced a list of Hauptmoduln, and conjectured the existence of an infinite-dimensional graded representation of M, whose graded traces Tg are the expansions of precisely the functions on their list.
The moonshine module and the proof
In 1980, A.O.L. Atkin, Paul Fong and Stephen D. Smith produced strong computational evidence that such a graded representation exists, by decomposing a large number of coefficients of J into representations of M. A graded representation whose graded dimension is J, called the moonshine module, was explicitly constructed by Igor Frenkel, James Lepowsky, and Arne Meurman, giving an effective solution to the McKay–Thompson conjecture, and they also determined the graded traces for all elements in the centralizer of an involution of M, partially settling the Conway–Norton conjecture. Furthermore, they showed that the vector space they constructed, called the Moonshine Module , has the additional structure of a vertex operator algebra, whose automorphism group is precisely M.
In 1985, the Atlas of Finite Groups was published by a group of mathematicians, including John Conway. The Atlas, which enumerates all sporadic groups, included "Moonshine" as a section in its list of notable properties of the monster group.
Borcherds proved the Conway–Norton conjecture for the Moonshine Module in 1992. He won the Fields Medal in 1998 in part for his solution of the conjecture; the award cited him "for his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie algebras, the proof of the Conway-Norton moonshine conjecture and the discovery of a new class of automorphic infinite products."
The moonshine module The Frenkel–Lepowsky–Meurman construction starts with two main tools:
The construction of a lattice vertex operator algebra VL for an even lattice L of rank n. In physical terms, this is the chiral algebra for a bosonic string compactified on a torus Rn/L. It can be described roughly as the tensor product of the group ring of L with the oscillator representation in n dimensions (which is itself isomorphic to a polynomial ring in countably infinitely many generators). For the case in question, one sets L to be the Leech lattice, which has rank 24. The orbifold construction. In physical terms, this describes a bosonic string propagating on a quotient orbifold. The construction of Frenkel–Lepowsky–Meurman was the first time orbifolds appeared in conformal field theory. Attached to the –1 involution of the Leech lattice, there is an involution h of VL, and an irreducible h-twisted VL-module, which inherits an involution lifting h. To get the Moonshine Module, one takes the fixed point subspace of h in the direct sum of VL and its twisted module.
Frenkel, Lepowsky, and Meurman then showed that the automorphism group of the moonshine module, as a vertex operator algebra, is M. Furthermore, they determined that the graded traces of elements in the subgroup 21+24.Co1 match the functions predicted by Conway and Norton.
Borcherds' proof Richard Borcherds's proof of the conjecture of Conway and Norton can be broken into the following major steps:
One begins with a vertex operator algebra V with an invariant bilinear form, an action of M by automorphisms, and with known decomposition of the homogeneous spaces of seven lowest degrees into irreducible M-representations. This was provided by Frenkel–Lepowsky–Meurman's construction and analysis of the Moonshine Module. A Lie algebra , called the monster Lie algebra, is constructed from V using a quantization functor. It is a generalized Kac–Moody Lie algebra with a monster action by automorphisms. Using the Goddard–Thorn "no-ghost" theorem from string theory, the root multiplicities are found to be coefficients of J. One uses the Koike–Norton–Zagier infinite product identity to construct a generalized Kac–Moody Lie algebra by generators and relations. The identity is proved using the fact that Hecke operators applied to J yield polynomials in J. By comparing root multiplicities, one finds that the two Lie algebras are isomorphic, and in particular, the Weyl denominator formula for is precisely the Koike–Norton–Zagier identity. Using Lie algebra homology and Adams operations, a twisted denominator identity is given for each element. These identities are related to the McKay–Thompson series Tg in much the same way that the Koike–Norton–Zagier identity is related to J. The twisted denominator identities imply recursion relations on the coefficients of Tg, and unpublished work of Koike showed that Conway and Norton's candidate functions satisfied these recursion relations. These relations are strong enough that one only needs to check that the first seven terms agree with the functions given by Conway and Norton. The lowest terms are given by the decomposition of the seven lowest degree homogeneous spaces given in the first step.
Thus, the proof is completed. Borcherds was later quoted as saying, "I was over the moon when I proved the moonshine conjecture", and "I sometimes wonder if this is the feeling you get when you take certain drugs. I don't actually know, as I have not tested this theory of mine."
More recent work has simplified and clarified the last steps of the proof. Jurisich found that the homology computation could be substantially shortened by replacing the usual triangular decomposition of the Monster Lie algebra with a decomposition into a sum of gl2 and two free Lie algebras. Cummins and Gannon showed that the recursion relations automatically imply the McKay-Thompson series are either Hauptmoduln or terminate after at most 3 terms, thus eliminating the need for computation at the last step.
Generalized moonshine
Conway and Norton suggested in their 1979 paper that perhaps moonshine is not limited to the monster, but that similar phenomena may be found for other groups. While Conway and Norton's claims were not very specific, computations by Larissa Queen in 1980 strongly suggested that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of irreducible representations of sporadic groups. In particular, she decomposed the coefficients of McKay-Thompson series into representations of subquotients of the Monster in the following cases:
T2B and T4A into representations of the Conway group Co0 T3B and T6B into representations of the Suzuki group 3.2.Suz T3C into representations of the Thompson group Th = F3