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In mathematics, ''monstrous moonshine'', or ''moonshine theory'', is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. It is now known that lying behind monstrous moonshine is a certain conformal field theory having the monster group as symmetries. The conjectures made by Conway and Norton were proved by Richard Borcherds in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. == History == In 1978, John McKay found that the first few terms in the Fourier expansion of ''j''(τ) , : with and ''τ'' as the half-period ratio 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. Let = 1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, ... then, : (Since there can be several linear relations between the ''r''n such as , the representation can 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. Conway and Norton computed the lower-order terms of such graded traces, now known as McKay–Thompson series ''T''''g'', and found that all of them appeared to be the expansions of Hauptmoduln. In other words, if ''G''''g'' is the subgroup of SL2(R) which fixes ''T''''g'', then the quotient of the upper half of the complex plane by ''G''''g'' is a sphere with a finite number of points removed, and furthermore, ''T''''g'' 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 ''T''''g'' are the expansions of precisely the functions on their list. In 1980, A. Oliver L. Atkin, Paul Fong and Stephen D. Smith, showed that such a graded representation exists, using computer calculation to decompose coefficients of ''j'' into representations of ''M'' up to a bound discovered by Thompson. A graded representation was explicitly constructed by Igor Frenkel, James Lepowsky, and Arne Meurman, giving an effective solution to the McKay–Thompson 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''. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Monstrous moonshine」の詳細全文を読む スポンサード リンク
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