Monstrous moonshine
Encyclopedia
In mathematics
, monstrous moonshine, or moonshine theory, 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
).
by John McKay, found that the Fourier expansion of j(τ) , with τ denoting the half-period ratio, could be expressed in terms of linear combination
s of the dimension
s of the irreducible representations of M :
where
, and
Conway and Norton formulated conjecture
s concerning the functions
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 Gg is the subgroup of SL2(R) which fixes
, then the quotient
of the upper half of the complex plane
by Gg is a sphere
with a finite number of points removed, corresponding to the cusp
s of Gg.
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 algebra
s 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.
-dimensional graded M-module
with
for all m, where
From this it follows that every element g of M acts on each Vm and has character value
which can be used to construct the 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
is the normalized main modular function for K.
, James Lepowsky
and Arne 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 orbifold
ed by a reflection. The resulting module is called the Monster module.
' proof of the conjecture of Conway and Norton can be broken into five major steps as follows:
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."
(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" (in the sense of being a crazy or foolish idea). 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
.
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 mathematician
s Jean-Pierre Serre
, Andrew Ogg
and John G. Thompson
; they studied the quotient
of the hyperbolic plane
by subgroup
s 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 if and only if
p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg heard about the Monster group later on, and noticed that these were precisely the prime factor
s of the size of M, he published a paper offering a bottle of Jack Daniel's
whiskey to anyone who could explain this fact.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
and Simon P. Norton
Simon P. Norton
Simon Phillips Norton is a mathematician in Cambridge, England, who works on finite simple groups. He constructed the Harada–Norton group, and in 1979 together with John Conway proved there is a connection between the Monster group and the j-function in number theory. They dubbed this monstrous...
in 1979, used to describe the (then totally unexpected) connection between the monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
M and modular functions (particularly, the j function
J-invariant
In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...
).
History
Specifically, Conway and Norton, following an initial observationby John McKay, found that the Fourier expansion of j(τ) , with τ denoting the half-period ratio, could be expressed in terms of linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
s of the dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s of the irreducible representations of M :
where
, andConway and Norton formulated conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
s concerning the functions
obtained by replacing the tracesTrace (linear algebra)
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 of A, i.e.,...
on the identity
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
by the traces on other elements g of M. The most striking part of these conjectures is that all these functions are genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
zero. In other words, if Gg is the subgroup of SL2(R) which fixes
, then the quotientQuotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of the upper half of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
by Gg is a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
with a finite number of points removed, corresponding to the cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
s of Gg.
It turns out that lying behind monstrous moonshine is a certain string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
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
Goddard-Thorn theorem
In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem is a theorem about certain vector spaces...
from string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
and the theory of vertex operator algebra
Vertex operator algebra
In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics...
s and generalized Kac-Moody superalgebras. Borcherds won the Fields medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, 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...
for his work, and more connections between M and the j-function were subsequently discovered.
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 infiniteInfinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
-dimensional graded M-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
with
for all m, whereFrom this it follows that every element g of M acts on each Vm and has character value
which can be used to construct the 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
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
zero subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...
K of PSL2(R), commensurable with the modular group Γ = PSL2(Z
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
), such that
is the normalized main modular function for K.The Monster module
It was subsequently shown by A. Oliver L. Atkin, Paul Fong and Stephen D. 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. Igor FrenkelIgor Frenkel
Igor Borisovich Frenkel is a mathematician at Yale University working in representation theory and mathematical physics.-Biography:...
, James Lepowsky
James Lepowsky
James Lepowsky is a professor of mathematics at Rutgers University, New Jersey. Previously he taught at Yale University. He received his Ph.D from M.I.T. in 1970 where his advisor was Bertram Kostant. Lepowsky graduated from Stuyvesant High School in 1961, 16 years after Kostant...
and Arne Meurman
Arne Meurman
Arne Meurman is a Swedish mathematician working on finite groups and vertex operator algebras. Currently he is a professor at Lund University....
explicitly constructed this representation using vertex operators in conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
, describing bosonic string theory
Bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s.In the early 1970s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory became the real focus...
compactified on a 24-dimensional torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
generated by the Leech lattice
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
, and orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
ed by a reflection. The resulting module is called the Monster module.
Borcherds' proof
Richard BorcherdsRichard Borcherds
Richard Ewen Borcherds is a British mathematician specializing in lattices, number theory, group theory, and infinite-dimensional algebras. He was awarded the Fields Medal in 1998.- Personal life :...
' proof of the conjecture of Conway and Norton can be broken into five major steps as follows:
- A vertex algebra V is constructed that is a graded algebraGraded algebraIn mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
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 algebraMonster vertex algebraThe monster vertex algebra is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R...
. - A Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is constructed from V using the Goddard–Thorn "no-ghost" theorem from string theoryString theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
; this is a generalized Kac-Moody Lie algebra. - A denominator identity for
is constructed that is related to the coefficients of 
. - A number of twisted denominator identities are constructed that are similarly related to the series
. - The denominator identities are used to determine the numbers cm, using Hecke operatorHecke operatorIn mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , 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....
s, Lie algebra homology and Adams operationAdams operationIn mathematics, an Adams operationis a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams...
s.
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."
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 (namely 196884) was precisely the dimension of the Griess algebraGriess algebra
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. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct M...
(and thus exactly one more than the degree of the smallest faithful complex representation of the Monster group), replied that this was "moonshine" (in the sense of being a crazy or foolish idea). Thus, the term not only refers to the Monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
M; it also refers to the perceived craziness of the intricate relationship between M and the theory of modular functions
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
.
However, "moonshine" is also a slang
Slang
Slang is the use of informal words and expressions that are not considered standard in the speaker's language or dialect but are considered more acceptable when used socially. Slang is often to be found in areas of the lexicon that refer to things considered taboo...
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 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, Andrew Ogg
Andrew Ogg
Andrew Pollard Ogg is an American mathematician, a professor emeritus of mathematics at the University of California, Berkeley....
and John G. Thompson
John G. Thompson
John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....
; they studied the quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of the hyperbolic plane
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
by subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...
s of SL2(R), particularly, the normalizer Γ0(p)+ of Γ0(p) in SL(2,R). They found that the Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
zero if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg heard about the Monster group later on, and noticed that these were precisely the prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
s of the size of M, he published a paper offering a bottle of Jack Daniel's
Jack Daniel's
Jack Daniel's is a brand of sour mash Tennessee whiskey that is among the world's best-selling liquors. It is known for its square bottles and black label. As of November, 2007, one blogger was claiming that it was the best-selling whiskey in the world. It is produced in Lynchburg, Tennessee by...
whiskey to anyone who could explain this fact.