Conjecture Monstrous Moonshine

In Mathematical, monstrous moonshine is an English term conceived by John Horton Conway and Simon P. Norton in 1979, used to describe connection (then completely unexpected) between the Groupe Monster M and the modular functions (particularly, the function '' J '').

Precisely, Conway and Norton, according to an initial observation of John McKay, found that the Développement of Fourier of $j\; (\backslash \; tau)\; \backslash ,$ (OEIS A000521, with $\backslash \; tau\; \backslash ,$ indicating the Ratio of half-period) could be expressed in linear terms of combinations of the Dimension S of the irreducible representations of M (OEIS A001379)

$j\; (\backslash \; tau)\; =\; \backslash \; frac\; \{1\}\; +\; 744\; +\; 196884\; \{Q\}\; +\; 21493760\; \{Q\}\; ^2\; +\; 864299970\; \{Q\}\; ^3\; +\; \backslash \; cdots$

where $\{Q\}\; =\; e^\; \{2\; \backslash \; pi\; I\; \backslash \; tau\}\; \backslash ,$ and

Conway and Norton formulated Conjecture S concerning the functions $j\_g\; (\{Q\})\; \backslash ,$ obtained by replacing the traces about the identity by the traces on other elements G of M . The part more seizing these conjectures is that all these functions are of kind zero. In other words, if $G\_g\; \backslash ,$ are the sub-group of SSL ₂ ($\backslash \; mathbb\; \{R\}$) which fixes $j\_g\; (\{Q\})\; \backslash ,$, then the quotient of the higher half-plane of the Plan complex by $G\_g\; \backslash ,$ is a Sphère with a number finished of point removed, corresponding to the parabolic forms of $G\_g\; \backslash ,$.

It proves that behind monstrous moonshine some are Théorie of the cords having the Monstre group like symmetries; the conjectures made by Conway and Norton were shown by Richard Ewen Borcherds in 1992 by using the theorem without phantom starting from the Théorie of the cords, of the theory of the algebras vertex and the generalized superalgèbres of Kac-Moody. Borcherds received the Medal Fields for its work, and more connections between M and the function J were later on discovered.

Formal versions of the conjectures of Conway and Norton

The first conjecture made by Conway and Norton was what one called the " conjecture moonshine" ; it establishes that there exists a M - module graduated dimension Infini E

$V\; =\; \backslash \; bigoplus\_\; \{m\; \backslash \; geq\; -1\}\; V\_m$

with $\backslash \; dim\; (V\_m)\; =\; C\; (m)$ for all m , where

$j\; (\{Q\})\; =\; \backslash \; sum\_\; \{m\; \backslash \; geq\; -1\}\; c\_m\; \{Q\}\; ^m.$

Of this, it follows that each element G of M acts on each V_m and has a Valeur of character

$\backslash \; chi\_m\; (G)\; =\; \backslash \; mathrm\; \{tr\}\; (G|\_\; \{V\_m\})$

who can be used to build the Série of McKay-Thompson of G :

$T\_g\; (\{Q\})\; =\; \backslash \; sum\_\; \{m\; \backslash \; geq\; -1\}\; \backslash \; chi\_m\; (G)\; \{Q\}\; ^m$.

The second conjecture of Conway and Norton then establishes that with V like above, for each element G of M , there exists a Sous-groupe of kind zero K of $PSL\_2\; (\backslash \; mathbb\; \{R\})\; \backslash ,$, commensurable with the modular group Γ:

$\backslash \; Gamma\; =\; PSL\_2\; (\backslash \; mathbb\; \{Z\})\; \backslash ,$, such as $T\_g\; (\{Q\})\; \backslash ,$ is the principal modular Fonction standardized for K .

The module Monster

It was shown later by A. Oliver L. Atkin, Paul Fong and Frederic L. Smith by using data-processing calculations that there exists indeed a graduated representation of infinite size of the Monstre group of which the series of McKay-Thompson are precisely the Hauptmodul S found by Conway and Norton, I.B. Frenkel, J. Lepowsky and A. Meurman built this representation explicitly by using the operators vertex. The resulting module is called the Module Monster.

The demonstration of Borcherds

The demonstration of Richard Ewen Borcherds of the conjecture of Conway and Norton can be separate in five major stages as what follows:

1. a Algèbre vertex V is built, i.e. a graduated Algèbre being able to carry out the representation moonshine on M , and it is checked that the Monstre module has a structure of invariant algebra vertex under the action of M . V is thus called the Algèbre vertex Monstre.

2. a Algebra of Dregs $\backslash \; mathcal\; \{M\}\; \backslash ,$ is built starting from V by using the theorem " without fantôme" of Goddard-Thorn starting from the Theory of the cords; this is a Algèbre of generalized Dregs Kac-Moody.
3. a denominator identity for $\backslash \; mathcal\; \{M\}$ is built, i.e. connected to the coefficients of $j\; (\{Q\})\; \backslash ,$.
4. a number of denominators identities twisted are built which are connected in a way similar to the series $T\_g\; (\{Q\})\; \backslash ,$.
5. the denominators identities are used to determine the numbers c_m , using the operators of Hecke, the Homologie of algebra of Dregs and the operations of Adams.

Thus, the demonstration is supplemented. Borcherds was quoted later like having said " I was on the Moon when I showed the conjecture moonshine (moonlight) " , and " I wonder sometimes if it is what one ressend when one takes certain drugs. I currently do not know it, as I did not test this theory on me. "

Why “monstrous moonshine”?

Term “monstrous moonshine” (that one can translate by monstrous bazaar) was invented by Conway, which, when John McKay says to him at the end of years 1970 that the coefficients of $\{Q\}$ (concretely 196.884) were precisely the dimension of the Algèbre of Griess (and thus exactly one moreover than the degree of the smallest faithful representation complexes Monstre group), answered that this was “moonshine” (within the meaning of insane “nutcase” or “ideas”). Thus, the term refers not only to the Groupe Monster M ; it refers also to the madness perceived concerning the complicated relation between M and the theory of the modular functions.

However, “moonshine” is also a word of Argot for a illegally distilled Whiskey, and in fact, the name can be explained in the light of this. The Monstre group was studied in the years 1970 by the Mathématicien S Fricke, Andrew Ogg and John G. Thompson; they studied the quotient hyperbolic Plan by the sub-groups of $SL\_2\; (\backslash \; mathbb\; \{R\})\; \backslash ,$, particularly, the Normalisateur $\backslash \; Gamma\_0\; (p)\; ^+\; \backslash ,$ of Γ ₀ ( p ) in SSL (2, R ). They found that the Surface of Riemann resulting by taking the quotient from the hyperbolic plan by $\backslash \; Gamma\_0\; (p)\; ^+\; \backslash ,$ is of kind zero If p is 2,3,5,7,11,13,17,19,23,29,31,41,47,59 or 71 (i.e., a Prime number super-singular), and when Ogg heard later of the Monstre group and foot-note which these numbers were precisely the factors first size of M , it prepared a paper offering a bottle of whiskey Jack Daniel' S to whoever could explain this fact.

References

• John Horton Conway and Simon P. Norton, Monstrous Moonshine , Bull. London Maths. Plowshare 11,308-339, 1979.

• I.B. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster , Pure and Applied Maths., vol. 134, Academic Near, 1988
• Richard Ewen Borcherds, Monstrous Moonshine and Monstrous Binds Superalgebras , Invent. Maths. 109,405-444, 1992, online
• Terry Gannon, Monstrous Moonshine: Twenty-five The first years , 2004, online
• Terry Gannon, Monstrous Moonshine and the Classification off 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 Farmhouse. ISBN 0-7382-0204-5 (Provides introductory reviews to applications in physics) .

External bonds

• Bibliography Moonshine

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