In mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function). Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Simon P. Norton is a mathematician in Cambridge, England, who works on finite simple groups. ... This page refers to the year 1979. ... In mathematics, the Monster group M is a group of order    246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ... In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...

Specifically, Conway and Norton, following an initial observation by John McKay, found that the Fourier expansion of j(τ) (OEIS A000521, with τ denoting the half-period ratio) could be expressed in terms of linear combinations of the dimensions of the irreducible representations of M (OEIS A001379) John McKay is mathematician at Concordia in Canada, known for his discovery of monstrous moonshine and for his joint construction of some sporadic simple groups. ... In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function (often taken to have period 2π — in a sense, the simplest case) as a sum of periodic functions of the form which are harmonics of ei x. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ... In mathematics, the half-period ratio τ of an elliptic function j is the ratio of the two half-periods ω1 and ω2 of j, where j is defined in such a way that See also Modular form Categories: Math stubs ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... :For other senses of this word, see dimension (disambiguation). ... In mathematics, the term irreducible is used in several ways. ...

where q = e^2πiτ, and

Conway and Norton formulated conjectures concerning the functions j_g(q) obtained by replacing the traces on the identity by the traces on other elements g of M. The most striking part of these conjectures is that all these functions are genus zero. In other words, if G_g is the subgroup of SL₂(R) which fixes j_g(q), then the quotient of the upper half of the complex plane by G_g is a sphere with a finite number of points removed, corresponding to the cusps of G_g. In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. ...

It turns out that lying behind monstrous moonshine is a certain string theory having the Monster group as symmetries; the conjectures made by Conway and Norton were proven by Richard Ewen Borcherds in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac-Moody superalgebras. Borcherds won the Fields medal for his work, and more connections between M and the j-function were subsequently discovered. Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles... Richard Ewen Borcherds (born November 29, 1959) is a mathematician specializing in group theory and Lie algebras. ... 1992 (MCMXCII) was a leap year starting on Wednesday. ... In mathematics, and in particular, in the mathematical background of string theory, the Goddard-Thorn theorem (also called the no-ghost theorem) is a theorem about certain vector spaces. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a generalized Kac-Moody algebra is a Lie algebra that is similar to a Kac-Moody algebra, except that it is allowed to have imaginary simple roots. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ...

Formal versions of Conway's and Norton's conjectures

The first conjecture made by Conway and Norton was the so-called "moonshine conjecture"; it states that there is an infinite-dimensional graded M-module The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

with for all m, where

From this it follows that every element g of M acts on each V_m and has character value In mathematics, the character of a group representation &#961; : G &#8594; GLn is the function &#967; : G -> C which sends g in G to the trace (the sum of the diagonal elements) of the matrix &#961;(g). ...

which can be used to construct the McKay-Thompson series of g:

The second conjecture of Conway and Norton then states that with V as above, for every element g of M, there is a genus zero subgroup K of PSL₂(R), commensurable with the modular group Γ = PSL₂(Z), such that T_g(q) is the normalised main modular function for K. In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, the modular group &#915; (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... The integers are commonly denoted by the above symbol. ... In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as H/&#915; where H is the upper half-plane in the complex numbers, and &#915; is a Fuchsian group acting on H, with &#915; a subgroup of the modular group of integral 2×2 matrices. ...

The Monster module

It was subsequently shown by A. Oliver L. Atkin, Paul Fong and Frederic L. Smith using computer calculation that there is indeed an infinite-dimensional graded representation of the Monster group whose McKay-Thompson series are precisely the Hauptmoduls found by Conway and Norton, and I. B. Frenkel, J. Lepowsky and A. Meurman explicitly constructed this representation using vertex operators in conformal field theory describing bosonic string theory compactified on a 24-dimensional torus generated by the Leech lattice and orbifolded by a reflection. The resulting module is called the Monster module. A. Oliver L. Atkin is a Professor Emeritus of mathematics at the University of Illinois at Chicago. ... In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as H/&#915; where H is the upper half-plane in the complex numbers, and &#915; is a Fuchsian group acting on H, with &#915; a subgroup of the modular group of integral 2×2 matrices. ... Igor Borisovich Frenkel (born April 22, 1952) is a mathematician working in mathematical physics. ... Please wikify (format) this article or section as suggested in the Guide to layout and the Manual of Style. ... Arne Meurman is a mathematician working on finite groups and vertex operators. ... This article needs to be cleaned up to conform to a higher standard of quality. ... A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ... Bosonic string theory is the original version of string theory, developed in the late 1960s. ... A torus. ... In mathematics, the Leech lattice is a lattice &#923; in R24 discovered John Leech ( 16 (1964), 657--682). ... In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ... The monster vertex algebra is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. ...

Borcherds' proof

Richard Ewen Borcherds' proof of the conjecture of Conway and Norton can be broken into five major steps as follows:

1. A vertex algebra V is constructed that is a graded algebra affording the moonshine representations on M, and it is verified that the monster module has a vertex algebra structure invariant under the action of M. V is thus called the Monster vertex algebra.
2. A Lie algebra is constructed from V using the Goddard-Thorn "no-ghost" theorem from string theory; this is a generalized Kac-Moody Lie algebra.
3. A denominator identity for is constructed that is related to the coefficients of j(q).
4. A number of twisted denominator identities are constructed that are similarly related to the series T_g(q).
5. The denominator identities are used to determine the numbers c_m, using Hecke operators, Lie algebra homology and Adams operations.

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." In mathematics, a vertex operator algebra (abbreviated: VOA) is a certain kind of algebra that plays a key part in conformal field theory and other fields of study in physics, and has also proven useful in purely mathematical contexts such as moonshine theory. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... The monster vertex algebra is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, and in particular, in the mathematical background of string theory, the Goddard-Thorn theorem (also called the no-ghost theorem) is a theorem about certain vector spaces. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles... In mathematics, a generalized Kac-Moody algebra is a Lie algebra that is similar to a Kac-Moody algebra, except that it is allowed to have imaginary simple roots. ... In mathematics, in particular in the theory of modular forms, a Hecke operator is a certain kind of averaging operator that plays a significant role in the structure of vector spaces of modular forms (and more general automorphic representations). ... In mathematics, an Adams operation &#968;k is a cohomology operation in K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. ...

Why "monstrous moonshine"?

The term "monstrous moonshine" was coined by Conway, who, when told by John McKay in the late 1970s that the coefficient of q (namely 196884) was precisely the dimension of the Griess algebra (and thus exactly one more than the degree of the smallest faithful complex representation of the Monster group), replied that this was "moonshine" (crazy or foolish ideas). Thus, the term not only refers to the Monster group M; it also refers to the perceived craziness of the intricate relationship between M and the theory of modular functions. John McKay is mathematician at Concordia in Canada, known for his discovery of monstrous moonshine and for his joint construction of some sporadic simple groups. ... The 1970s decade refers to the years from 1970 to 1979, inclusive. ... In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group. ... In mathematics, the Monster group M is a group of order    246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ... In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ...

However, "moonshine" is also a slang word for illegally distilled whiskey, and in fact, the name may be explained in this light as well. The Monster group was investigated in the 1970s by mathematicians Fricke, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL₂(R), particularly, the normalizer Γ₀(p)⁺ of 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 iff p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71 (that is, a supersingular prime), and when Ogg heard about the Monster group later on and noticed that these were precisely the prime factors of the size of M, he wrote up a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact. Slang is the use of highly informal words and expressions that are not considered standard in the speakers dialect or language. ... Whisky (or whiskey) is an alcoholic beverage distilled from grain, often including malt, which has then been aged in wooden barrels. ... The 1970s decade refers to the years from 1970 to 1979, inclusive. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... John Griggs Thompson (born 13 Oct 1932) is a mathematician noted for his work in the field of finite groups. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Pearson surface, is a one-dimensional complex manifold. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In mathematics, a supersingular prime is a certain kind of prime number. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... Statue of Jack Daniel at the Distillery in November 2004. ...

References

• John Horton Conway and Simon P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11, 308–339, 1979.
• I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, 1988
• Richard Ewen Borcherds, Monstrous Moonshine and Monstrous Lie Superalgebras, Invent. Math. 109, 405–444, 1992, online
• Terry Gannon, Monstrous Moonshine: The first twenty-five years, 2004, online
• Terry Gannon, Monstrous Moonshine and the Classification of Conformal Field Theories, reprinted in Conformal Field Theory, New Non-Perturbative Methods in String and Field Theory, (2000) Yavuz Nutku, Cihan Saclioglu, Teoman Turgut, eds. Perseus Publishing, Cambridge Mass. ISBN 0-7382-0204-5 (Provides introductory reviews to applications in physics).

External links

• Moonshine bibliography