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| In [[mathematics]], '''''monstrous moonshine''''', or '''''moonshine theory''''', is a term devised by [[John Horton Conway|John Conway]] and [[Simon P. Norton]] in 1979, used to describe the unexpected connection between the [[monster group]] ''M'' and [[modular function]]s, in particular, the [[j-invariant|''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 [[Goddard–Thorn theorem|no-ghost theorem]] from [[string theory]] and the theory of [[vertex operator algebra]]s and [[generalized Kac–Moody algebra]]s.
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| == History ==
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| In 1978, [[John McKay (mathematics)|John McKay]] found that the first few terms in the [[Fourier expansion]] of ''j''(τ) {{OEIS|A000521}}, with τ denoting the [[half-period ratio]], could be expressed in terms of [[linear combination]]s of the [[dimension]]s of the [[irreducible representation]]s of ''M'' {{OEIS|A001379}} with very small natural coefficients:
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| :<math>j(\tau) = \frac{1}{{q}} + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + \cdots</math>
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| where <math>{q} = e^{2\pi i\tau}</math>, and
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| :<math>
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| \begin{align}
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| 1 & = 1 \\
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| 196884 & = 196883 + 1 \\
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| 21493760 & = 21296876 + 196883 + 1 \\
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| 864299970 & = 842609326 + 21296876 + 2\cdot 196883 + 2\cdot 1 \\
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| 20245856256&=18538750076+2\cdot842609326+21296876+3\cdot196883+3\cdot1\\
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| &=19360062527+842609326+2\cdot21296876+3\cdot196883+2\cdot1\\
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| 333202640600 & =293553734298+2\cdot 18538750076+3\cdot 842609326+2\cdot 21296876+5\cdot 196883+5\cdot 1\\
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| & =293553734298+19360062527+ 18538750076+2\cdot 842609326+3\cdot 21296876+5\cdot 196883+4\cdot 1\\
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| 4252023300096 & =3879214937598+293553734298+4\cdot 18538750076+6\cdot842609326+2\cdot21296876+7\cdot196883+7\cdot1 \\
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| & {}\,\,\, \vdots
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| \end{align}
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| </math>
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| McKay viewed this as evidence that there is a naturally occurring infinite-dimensional [[Graded_vector_space|graded representation]] of ''M'', whose [[Hilbert–Poincaré_series|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 (linear algebra)|trace]] of the [[identity element]], the graded traces of nontrivial elements ''g'' of ''M'' on such a representation may be interesting as well.
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| Conway and Norton computed the lower-order terms of such graded traces, now known as [[McKay–Thompson series]] ''T''<sub>''g''</sub>, and found that all of them appeared to be the expansions of [[Modular curve#Genus zero|Hauptmoduln]]. In other words, if ''G''<sub>''g''</sub> is the subgroup of [[SL2(R)|SL<sub>2</sub>('''R''')]] which fixes ''T''<sub>''g''</sub>, then the [[quotient group|quotient]] of the [[upper half-plane|upper half]] of the [[complex plane]] by ''G''<sub>''g''</sub> is a [[sphere]] with a finite number of points removed, and furthermore, ''T''<sub>''g''</sub> generates the [[field (mathematics)|field]] of [[meromorphic function]]s on this sphere.
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| 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''<sub>''g''</sub> are the [[Fourier expansion|expansion]]s of precisely the functions on their list.
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| 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 <math>V^\natural</math>, has the additional structure of a [[vertex operator algebra]], whose [[automorphism group]] is precisely ''M''.
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| 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.
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| == The Monster module ==
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| The Frenkel–Lepowsky–Meurman construction uses two main tools:
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| # The construction of a lattice vertex operator algebra ''V''<sub>''L''</sub> for an even [[lattice (group)|lattice]] ''L'' of rank ''n''. In physical terms, this is the [[chiral algebra]] for a [[bosonic string]] [[Compactification_(physics)|compactified]] on a [[torus]] '''R'''<sup>''n''</sup>/''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 infinite]]ly many [[generator matrix|generators]]). For the case in question, one sets ''L'' to be the [[Leech lattice]], which has rank 24.
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| # The [[orbifold]] construction. In physical terms, this describes a bosonic string propagating on a [[orbifold|quotient orbifold]]. The construction of Frenkel–Lepowsky–Meurman was the first time orbifolds appeared in [[conformal field theory]]. Attached to the [[Involution_(mathematics)|–1 involution]] of the [[Leech lattice]], there is an involution ''h'' of ''V''<sub>''L''</sub>, and an irreducible ''h''-[[twist (mathematics)|twist]]ed ''V''<sub>''L''</sub>module, which inherits an involution lifting ''h''. To get the Moonshine Module, one takes the [[Fixed_point_(mathematics)|fixed point subspace]] of ''h'' in the direct sum of ''V''<sub>''L''</sub> and its [[vertex operator algebra|twisted module]].
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| Frenkel, Lepowsky, and Meurman showed that the automorphism group of the moonshine module, as a vertex operator algebra, is ''M'', and they showed that its graded dimension gives the Fourier expansion of ''j'' ({{harvtxt|Frenkel|Lepowsky|Meurman|1988}}).
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| == Borcherds' proof ==
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| [[Richard Borcherds]]' proof of the conjecture of Conway and Norton can be broken into the following major steps:
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| # One begins with a vertex operator algebra ''V'', with an action of ''M'' by automorphisms, and with graded dimension ''j''. This was provided by the Moonshine Module, also called the monster vertex algebra or monster VOA.
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| # A [[Lie algebra]] <math>\mathfrak{m}</math>, called the [[monster Lie algebra]], is constructed from ''V'' using a quantization functor. It is a [[Generalized Kac–Moody algebra|generalized Kac–Moody Lie algebra]] with a monster action by automorphisms. Using the [[Goddard–Thorn theorem|Goddard–Thorn "no-ghost" theorem]] from [[string theory]], the root multiplicities are found to be coefficients of ''j''.
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| # 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 operator]]s applied to ''j'' yield polynomials in ''j''.
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| # By comparing root multiplicities, one finds that the two Lie algebras are isomorphic, and in particular, the [[Weyl denominator formula]] for <math>\mathfrak{m}</math> is precisely the Koike–Norton–Zagier identity.
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| # Using [[Lie algebra homology]] and [[Adams operation]]s, a twisted denominator identity is given for each element. These identities are related to the McKay–Thompson series ''T''<sub>g</sub> in much the same way that the Koike–Norton–Zagier identity is related to ''j''.
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| # The twisted denominator identities imply recursion relations on the coefficients of ''T''<sub>g</sub>. 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.
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| Thus, the proof is completed ({{harvtxt|Borcherds|1992}}). 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.''"{{cn|date=August 2013}}
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| == Generalized Moonshine ==
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| 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. In 1980, Larissa Queen found that one can in fact construct the expansions of many Hauptmoduln of level greater than one from simple combinations of dimensions of [[sporadic group]]s, such as the [[Baby monster]]. For example, the smallest [[faithful representation]] of the Baby monster has dimension 4371, while the Hauptmodul for Γ<sub>0</sub>(2)<sup>+</sup> has expansion <math>q^{-1} + 4372q + \cdots</math>.
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| In 1987, Norton combined Queen's results with his own computations to formulate the Generalized Moonshine conjecture. This conjecture asserts that there is a rule that assigns to each element ''g'' of the monster, a graded vector space ''V''(''g''), and to each commuting pair of elements (''g'',''h'') a [[holomorphic function]] ''f''(''g'',''h'',τ) on the [[upper half plane]], such that:
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| # Each ''V''(''g'') is a graded projective representation of the [[centralizer]] of ''g'' in ''M''.
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| # Each ''f''(''g'',''h'',τ) is either a constant function, or a Hauptmodul.
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| # Each ''f''(''g'',''h'',τ) is invariant under simultaneous [[conjugation (group theory)|conjugation]] of ''g'' and ''h'' in ''M''.
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| # For each (''g'',''h''), there is a lift of ''h'' to a [[linear transformation]] on ''V''(''g''), such that the expansion of ''f''(''g'',''h'',τ) is given by the graded trace.
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| # For any <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z})</math>, <math>f(g,h,\frac{a\tau+b}{c\tau+d})</math> is proportional to <math>f(g^a h^c, g^b h^d, \tau)</math>.
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| # ''f''(''g'',''h'',τ) is proportional to ''j'' if and only if ''g'' = ''h'' = 1.
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| This is a generalization of the Conway–Norton conjecture, because Borcherds's theorem concerns the case where ''g'' is set to the identity. To date, this conjecture is still open.
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| Like the Conway–Norton conjecture, Generalized Moonshine also has an interpretation in physics, proposed by Dixon–Ginsparg–Harvey in 1988 ({{harvtxt|Dixon|Ginsparg|Harvey|1989}}). They interpreted the vector spaces ''V''(''g'') as twisted sectors of a conformal field theory with monster symmetry, and interpreted the functions ''f''(''g'',''h'',τ) as [[Genus_(mathematics)|genus]] one [[partition function (mathematics)|partition function]]s, where one forms a torus by gluing along twisted boundary conditions. In mathematical language, the twisted sectors are irreducible twisted modules, and the partition functions are assigned to elliptic curves with principal monster bundles, whose isomorphism type is described by [[monodromy]] along a [[Generating set of a group|basis]] of [[1-cycle]]s, i.e., a pair of commuting elements.
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| == Conjectured relationship with quantum gravity ==
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| In 2007, [[Edward Witten|E. Witten]] suggested that [[AdS/CFT correspondence]] yields a duality between pure quantum gravity in (2+1)-dimensional [[Anti-de Sitter space|anti de Sitter space]] and extremal holomorphic CFTs. Pure gravity in 2+1 dimensions has no local degrees of freedom, but when the cosmological constant is negative, there is nontrivial content in the theory, due to the existence of [[BTZ black hole]] solutions. Extremal CFTs, introduced by G. Höhn, are distinguished by a lack of Virasoro primary fields in low energy, and the moonshine module is one example.
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| Under Witten's proposal ({{harvtxt|Witten|2007}}), gravity in AdS space with maximally negative cosmological constant is AdS/CFT dual to a holomorphic CFT with central charge ''c=24'', and the partition function of the CFT is precisely ''j-744'', i.e., the graded character of the moonshine module. By assuming Frenkel-Lepowsky-Meurman's conjecture that moonshine module is the unique holomorphic VOA with central charge 24 and character ''j-744'', Witten concluded that pure gravity with maximally negative cosmological constant is dual to the Monster CFT. Part of Witten's proposal is that Virasoro primary fields are dual to black-hole-creating operators, and as a consistency check, he found that in the large-mass limit, the [[Black_hole_thermodynamics|Beckenstein-Hawking]] semiclassical entropy estimate for a given black hole mass agrees with the logarithm of the corresponding Virasoro primary multiplicity in the moonshine module. In the low-mass regime, there is a small quantum correction to the entropy, e.g., the lowest energy primary fields yield log(196883) ~ 12.19, while the Beckenstein-Hawking estimate gives 4π ~ 12.57.
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| Later work has refined Witten's proposal. Witten had speculated that the extremal CFTs with larger cosmological constant may have monster symmetry much like the minimal case, but this was quickly ruled out by independent work of Gaiotto and Höhn. Work by Witten and Maloney ({{harvtxt|Maloney|Witten|2007}}) suggested that pure quantum gravity may not satisfy some consistency checks related to its partition function, unless some subtle properties of complex saddles work out favorably. However, Li-Song-Strominger ({{harvtxt|Li|Song|Strominger|2008}}) have suggested that a chiral quantum gravity theory proposed by Manschot in 2007 may have better stability properties, while being dual to the chiral part of the monster CFT, i.e., the monster vertex algebra. Duncan-Frenkel ({{harvtxt|Duncan|Frenkel|2009}}) produced additional evidence for this duality by using [[Rademacher sum]]s to produce the McKay-Thompson series as 2+1 dimensional gravity partition functions by a regularized sum over global torus-isogeny geometries. Furthermore, they conjectured the existence of a family of twisted chiral gravity theories parametrized by elements of the monster, suggesting a connection with generalized moonshine and gravitational instanton sums. At present, all of these ideas are still rather speculative, in part because 3d quantum gravity does not have a rigorous mathematical foundation.
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| == Mathieu Moonshine ==
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| In 2010, Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa observed that the elliptic genus of a [[K3 surface]] can be decomposed into characters of the ''N''=(4,4) [[superconformal algebra]], such that the multiplicities of [[Super_Virasoro_algebra|massive state]]s appear to be simple combinations of irreducible representations of the [[Mathieu group M24]]. This suggests that there is a sigma-model [[conformal field theory]] with K3 target that carries M24 symmetry. However, by the Mukai–Kondo classification, there is no [[faithful action]] of this group on any K3 surface by [[Symplectomorphism|symplectic automorphism]]s, and by work of Gaberdiel–Hohenegger–Volpato, there is no faithful action on any K3 sigma-model conformal field theory, so the appearance of an action on the underlying [[Hilbert space]] is still a mystery.
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| By analogy with McKay–Thompson series, Cheng suggested that both the [[multiplicity function]]s and the graded traces of nontrivial elements of M24 form [[Mock modular form]]s. In 2012, Gannon proved that all but the first of the multiplicities are non-negative [[Linear_combination|integral combination]]s of representations of M24, and Gaberdiel–Persson–Ronellenfitsch–Volpato computed all analogues of generalized moonshine functions, strongly suggesting that some analogue of a holomorphic conformal field theory lies behind Mathieu moonshine. Also in 2012, Cheng, Duncan, and [[Jeffrey A. Harvey|Harvey]] amassed numerical evidence of an [[umbral moonshine]] phenomenon where families of mock modular forms appear to be attached to Niemeier lattices. The special case of the ''A''<sub>1</sub><sup>24</sup> lattice yields Mathieu Moonshine, but in general the phenomenon does not yet have an interpretation in terms of geometry.
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| == Why "monstrous moonshine"? ==
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| The term "monstrous moonshine" was coined by Conway, who, when told by [[John McKay (mathematics)|John McKay]] in the late 1970s that the coefficient of <math>{q}</math> (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 "[[wikt:moonshine|moonshine]]" (in the sense of being a crazy or foolish idea).<ref>[http://www.worldwidewords.org/topicalwords/tw-moo1.htm World Wide Words: Moonshine]</ref> 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 function]]s.
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| However, "moonshine" is also a [[slang]] word for illegally distilled [[whiskey]], and in fact the name may be explained in this light as well.{{cn|date=August 2013}} The Monster group was investigated in the 1970s by [[mathematician]]s [[Jean-Pierre Serre]], [[Andrew Ogg]] and [[John G. Thompson]]; they studied the [[quotient group|quotient]] of the [[Hyperbolic space|hyperbolic plane]] by [[subgroup]]s of SL<sub>2</sub>('''R'''), particularly, the [[normalizer]] Γ<sub>0</sub>(''p'')<sup>+</sup> of [[modular group Gamma0|Γ<sub>0</sub>]](''p'') in SL(2,'''R'''). They found that the [[Riemann surface]] resulting from taking the quotient of the [[hyperbolic plane]] by Γ<sub>0</sub>(''p'')<sup>+</sup> has [[genus (mathematics)|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 ({{harvtxt|Ogg|1974}}).
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| == Notes ==
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| {{reflist}}
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| == References ==
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| {{refbegin}}
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| * [[John Horton Conway]] and [[Simon P. Norton]], ''Monstrous Moonshine'', Bull. London Math. Soc. 11, 308–339, 1979.
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| *{{Citation | first1 = I. | last1 = Frenkel |first2 = J. | last2 = Lepowsky | first3 = A. | last3 = Meurman | title = Vertex Operator Algebras and the Monster | journal=Pure and Applied Math. | publisher=Academic Press | year = 1988 | volume = 134 }}
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| *{{Citation | first = Richard | last = Borcherds | title = Monstrous Moonshine and Monstrous Lie Superalgebras | journal=Invent. Math. | year = 1992 | volume = 109 |page=405–444 | url=http://math.berkeley.edu/~reb/papers/ }}
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| * Terry Gannon, ''Monstrous Moonshine: The first twenty-five years'', 2004, [http://arxiv.org/abs/math.QA/0402345 online]
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| * 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)''.
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| *{{Citation | first = Terry | last = Gannon | title = Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics | year = 2006 | isbn = 0-521-83531-3}}
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| *{{Citation | first1 = L. | last1 = Dixon |first2 = P. | last2 = Ginsparg | first3 = J. | last3 = Harvey | title = Beauty and the Beast: superconformal symmetry in a Monster module | journal=Comm. Math. Phys. | year = 1989 | volume = 119 | page = 221–241 }}
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| *{{citation | last = Witten | first = Edward | authorlink = Edward Witten | date = 22 June 2007 | year= 2007 |title = Three-Dimensional Gravity Revisited |url=http://arxiv.org/pdf/0706.3359.pdf }}
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| *{{citation | last1 = Maloney | first1 = Alexander | last2 = Witten | first2 = Edward | date = 2 December 2007 | year= 2007 |title = Quantum Gravity Partition Functions In Three Dimensions | url=http://arxiv.org/pdf/0712.0155.pdf }}
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| *{{citation | last1 = Li | first1 = Wei | last2 = Song | first2 = Wei | last3 = Strominger | first3 = Andrew | date = 21 July 2008 | year= 2008 | title = CHIRAL GRAVITY IN THREE DIMENSIONS | url=http://arxiv.org/pdf/0801.4566.pdf }}
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| *{{citation | last1 = Duncan | first1 = John F. R. | last2 = Frenkel | first2 = Igor B. | date = 12 April 2012 |year=2009 | title=Rademacher sums, moonshine and gravity | url=http://arxiv.org/pdf/0907.4529.pdf }}
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| *{{Citation |first1=Alexander |last1=Maloney |first2=Wei |last2=Song |first3=Andrew |last3=Strominger |title=Chiral gravity, log gravity, and extremal CFT |journal=Phys. Rev. D |volume=81 |issue= |pages= |year=2010|url=http://arxiv.org/pdf/0903.4573.pdf}}
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| * Koichiro Harada, ''Monster'', Iwanami Pub. (1999) ISBN 4-00-006055-4, ''(The first book about the Monster Group written in Japanese)''.
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| * [[Mark Ronan]], ''Symmetry and the Monster'', Oxford University Press, 2006. ISBN 978-0-19-280723-6 ''(Concise introduction for the lay reader)''.
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| * [[Marcus Du Sautoy]], ''Finding Moonshine'', ''A Mathematician's Journey Through Symmetry''. Fourth Estate, 2008 ISBN 0-00-721461-8, ISBN 978-0-00-721461-7
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| *{{citation | last = Ogg | first = Andrew | authorlink = Andrew P. Ogg |year= 1974 |title = Automorphismes de courbes modulaires | journal=Seminaire Delange-Pisot-Poitou. Theorie des nombres, tome 16, no. 1 (1974–1975), exp. no. 7 |url=http://archive.numdam.org/ARCHIVE/SDPP/SDPP_1974-1975__16_1/SDPP_1974-1975__16_1_A4_0/SDPP_1974-1975__16_1_A4_0.pdf}}
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| {{refend}}
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| == External links ==
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| * {{wayback|url=http://web.archive.org/web/20061205032152/http://cicma.mathstat.concordia.ca/faculty/cummins/moonshine.refs.html|title=Moonshine Bibliography}}
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| [[Category:Moonshine theory|*]]
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| [[Category:Group theory]]
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| [[Category:Words coined in the 1970s]]
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