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Inmathematics,monstrous moonshine, ormoonshine theory, is the unexpected connection between themonster groupM andmodular functions, in particular thej function. The initial numerical observation was made byJohn McKay in 1978, and the phrase was coined byJohn Conway andSimon P. Norton in 1979.^[1][2][3]

The monstrous moonshine is now known to be underlain by avertex operator algebra called themoonshine module (or monster vertex algebra) constructed byIgor Frenkel,James Lepowsky, andArne Meurman in 1988, which has the monster group as its group ofsymmetries. This vertex operator algebra is commonly interpreted as a structure underlying atwo-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven byRichard Borcherds for the moonshine module in 1992 using theno-ghost theorem fromstring theory and the theory ofvertex operator algebras andgeneralized Kac–Moody algebras.

In 1978,John McKay found that the first few terms in theFourier expansion of the normalizedJ-invariant (sequenceA014708 in theOEIS) could be expressed in terms oflinear combinations of thedimensions of theirreducible representations${\displaystyle r_n}$ of the monster groupM (sequenceA001379 in theOEIS) withsmall non-negative coefficients. The J-invariant is$${\displaystyle J(\tau )=\frac{1}{q}+744+196884q+21493760q^2+864299970q^3+20245856256q^4+\cdots }$$with${\displaystyle q=e^{2\pi i\tau }}$ andτ as thehalf-period ratio, and theM expressions, letting${\displaystyle r_n}$ = 1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, ..., are

The LHS are the coefficients of${\displaystyle j(\tau )}$, while in the RHS the integers${\displaystyle r_n}$ are the dimensions ofirreducible representations of the monster groupM. (Since there can be several linear relations between the${\displaystyle r_n}$ such as${\displaystyle r_1-r_3+r_4+r_5-r_6=0}$, the representation may be in more than one way.)

McKay viewed this as evidence that there is a naturally occurring infinite-dimensionalgraded representation ofM, whosegraded dimension is given by the coefficients ofJ, and whose lower-weight pieces decompose into irreducible representations as above. After he informedJohn G. Thompson of this observation, Thompson suggested that because the graded dimension is just the gradedtrace of theidentity element, the graded traces of nontrivial elementsg ofM 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 seriesT_g, and found that all of them appeared to be the expansions ofHauptmoduln. In other words, ifG_g is the subgroup ofSL₂(R) which fixesT_g, then thequotient of theupper half of thecomplex plane byG_g is asphere with a finite number of points removed, and furthermore,T_g generates thefield ofmeromorphic functions on this sphere.

Based on their computations, Conway and Norton produced a list ofHauptmoduln, and conjectured the existence of an infinite dimensional graded representation ofM, whose graded tracesT_g are theexpansions of precisely the functions on their list.

In 1980,A.O.L. Atkin, Paul Fong and Stephen D. Smith produced strong computational evidence that such a graded representation exists, by decomposing a large number of coefficients ofJ into representations ofM. A graded representation whose graded dimension isJ, called the moonshine module, was explicitly constructed byIgor Frenkel,James Lepowsky, andArne Meurman, giving an effective solution to the McKay–Thompson conjecture, and they also determined the graded traces for all elements in the centralizer of an involution ofM, partially settling the Conway–Norton conjecture. Furthermore, they showed that thevector space they constructed, called the Moonshine Module${\displaystyle V^{\mathrm{♮}}}$, has the additional structure of avertex operator algebra, whoseautomorphism group is preciselyM.

In 1985, theAtlas of Finite Groups was published by a group of mathematicians, includingJohn Conway. The Atlas, which enumerates allsporadic groups, included "Moonshine" as a section in its list of notable properties of themonster group.^[4]

Borcherds proved the Conway–Norton conjecture for the Moonshine Module in 1992. He won theFields Medal in 1998 in part for his solution of the conjecture.

The Frenkel–Lepowsky–Meurman construction starts with two main tools:

1. The construction of a lattice vertex operator algebraV_L for an evenlatticeL of rankn. In physical terms, this is thechiral algebra for abosonic stringcompactified on atorusRⁿ/L. It can be described roughly as thetensor product of thegroup ring ofL with the oscillator representation inn dimensions (which is itself isomorphic to apolynomial ring incountably infinitely manygenerators). For the case in question, one setsL to be theLeech lattice, which has rank 24.
2. Theorbifold construction. In physical terms, this describes a bosonic string propagating on aquotient orbifold. The construction of Frenkel–Lepowsky–Meurman was the first time orbifolds appeared inconformal field theory. Attached to the–1 involution of theLeech lattice, there is an involutionh ofV_L, and an irreducibleh-twistedV_L-module, which inherits an involution liftingh. To get the Moonshine Module, one takes thefixed point subspace ofh in the direct sum ofV_L and itstwisted module.

Frenkel, Lepowsky, and Meurman then showed that the automorphism group of the moonshine module, as a vertex operator algebra, isM. Furthermore, they determined that the graded traces of elements in the subgroup 2¹⁺²⁴.Co₁ match the functions predicted by Conway and Norton.^[5]

Richard Borcherds' proof of the conjecture of Conway and Norton can be broken into the following major steps:

1. One begins with a vertex operator algebraV with an invariantbilinear form, an action ofM by automorphisms, and with known decomposition of the homogeneous spaces of seven lowest degrees into irreducibleM-representations. This was provided by Frenkel–Lepowsky–Meurman's construction and analysis of the Moonshine Module.
2. ALie algebra${\displaystyle \mathfrak{m}}$, called themonster Lie algebra, is constructed fromV using a quantization functor. It is ageneralized Kac–Moody Lie algebra with a monster action by automorphisms. Using theGoddard–Thorn "no-ghost" theorem fromstring theory, the root multiplicities are found to be coefficients ofJ.
3. 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 thatHecke operators applied toJ yield polynomials inJ.
4. By comparing root multiplicities, one finds that the two Lie algebras are isomorphic, and in particular, theWeyl denominator formula for${\displaystyle \mathfrak{m}}$ is precisely the Koike–Norton–Zagier identity.
5. UsingLie algebra homology andAdams operations, a twisted denominator identity is given for each element. These identities are related to the McKay–Thompson seriesT_g in much the same way that the Koike–Norton–Zagier identity is related toJ.
6. The twisted denominator identities imply recursion relations on the coefficients ofT_g, and unpublished work of Koike showed that Conway and Norton's candidate functions satisfied these recursion relations. 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. The lowest terms are given by the decomposition of the seven lowest degree homogeneous spaces given in the first step.

Thus, the proof is completed.^[6] 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."^[7]

More recent work has simplified and clarified the last steps of the proof. Jurisich found that the homology computation could be substantially shortened by replacing the usual triangular decomposition of the Monster Lie algebra with a decomposition into a sum ofgl₂ and two free Lie algebras.^[8][9] Cummins and Gannon showed that the recursion relations automatically imply the McKay-Thompson series are either Hauptmoduln or terminate after at most 3 terms, thus eliminating the need for computation at the last step.

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.^[a] While Conway and Norton's claims were not very specific, computations by Larissa Queen in 1980 strongly suggested that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of irreducible representations ofsporadic groups. In particular, she decomposed the coefficients of McKay-Thompson series into representations of subquotients of the Monster in the following cases:

• T_2B andT_4A into representations of theConway group Co₀
• T_3B andT_6B into representations of theSuzuki group 3.2.Suz
• T_3C into representations of theThompson groupTh = F₃
• T_5A into representations of theHarada–Norton groupHN = F₅
• T_5B andT_10D into representations of theHall–Janko group 2.HJ
• T_7A into representations of theHeld groupHe = F₇
• T_7B andT_14C into representations of 2.A₇
• T_11A into representations of theMathieu group 2.M₁₂

Queen found that the traces of non-identity elements also yieldedq-expansions of Hauptmoduln, some of which were not McKay–Thompson series from the Monster. 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 elementg of the monster, a graded vector spaceV(g), and to each commuting pair of elements (g,h) aholomorphic functionf(g,h, τ) on theupper half-plane, such that:

1. EachV(g) is a gradedprojective representation of thecentralizer ofg inM.
2. Eachf(g,h, τ) is either a constant function, or a Hauptmodul.
3. Eachf(g,h, τ) is invariant under simultaneousconjugation ofg andh inM, up to a scalar ambiguity.
4. For each (g,h), there is a lift ofh to alinear transformation onV(g), such that the expansion off(g,h, τ) is given by the graded trace.
5. For any,${\displaystyle f(g,h,\frac{a\tau +b}{c\tau +d})}$ is proportional to${\displaystyle f(g^a{h}^c,g^b{h}^d,\tau )}$.
6. f(g,h,τ) is proportional toJ if and only ifg =h = 1.

This is a generalization of the Conway–Norton conjecture, because Borcherds's theorem concerns the case whereg is set to the identity.

Like the Conway–Norton conjecture, Generalized Moonshine also has an interpretation in physics, proposed by Dixon–Ginsparg–Harvey in 1988.^[10] They interpreted the vector spacesV(g) as twisted sectors of a conformal field theory with monster symmetry, and interpreted the functionsf(g,h, τ) asgenus onepartition functions, 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 bymonodromy along abasis of1-cycles, i.e., a pair of commuting elements.

In the early 1990s, the group theorist A. J. E. Ryba discovered remarkable similarities between parts of thecharacter table of the monster, andBrauer characters of certain subgroups. In particular, for an elementg of primeorderp in the monster, many irreducible characters of an element of orderkp whosekth power isg are simple combinations of Brauer characters for an element of orderk in the centralizer ofg. This was numerical evidence for a phenomenon similar to monstrous moonshine, but for representations inpositive characteristic. In particular, Ryba conjectured in 1994 that for each prime factorp in the order of the monster, there exists a graded vertex algebra over thefinite fieldF_p with an action of the centralizer of an orderp elementg, such that the graded Brauer character of anyp-regular automorphismh is equal to the McKay-Thompson series forgh.^[11]

In 1996, Borcherds and Ryba reinterpreted the conjecture as a statement aboutTate cohomology of a self-dual integral form of${\displaystyle V^{\mathrm{♮}}}$. This integral form was not known to exist, but they constructed a self-dual form overZ[1/2], which allowed them to work with odd primesp. The Tate cohomology for an element of prime order naturally has the structure of a super vertex algebra overF_p, and they broke up the problem into an easy step equating graded Brauer super-trace with the McKay-Thompson series, and a hard step showing that Tate cohomology vanishes in odd degree. They proved the vanishing statement for small odd primes, by transferring a vanishing result from the Leech lattice.^[12] In 1998, Borcherds showed that vanishing holds for the remaining odd primes, using a combination ofHodge theory and an integral refinement of theno-ghost theorem.^[13][14]

The case of order 2 requires the existence of a form of${\displaystyle V^{\mathrm{♮}}}$ over a 2-adic ring, i.e., a construction that does not divide by 2, and this was not known to exist at the time. There remain many additional unanswered questions, such as how Ryba's conjecture should generalize to Tate cohomology of composite order elements, and the nature of any connections to generalized moonshine and other moonshine phenomena.

In 2007,E. Witten suggested thatAdS/CFT correspondence yields a duality between pure quantum gravity in (2 + 1)-dimensionalanti de Sitter space and extremal holomorphic CFTs. Pure gravity in 2 + 1 dimensions has no local degrees of freedom, but when thecosmological constant is negative, there is nontrivial content in the theory, due to the existence ofBTZ 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.

Under Witten's proposal, gravity in AdS space with maximally negative cosmological constant is AdS/CFT dual to a holomorphic CFT with central chargec=24, and the partition function of the CFT is preciselyj-744, i.e., the graded character of the moonshine module.^[15] By assuming Frenkel-Lepowsky-Meurman's conjecture that moonshine module is the unique holomorphic VOA withcentral charge 24 and characterj-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, theBekenstein-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 ln(196883) ~ 12.19, while the Bekenstein–Hawking estimate gives 4π ~ 12.57.

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 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.^[16] However, Li–Song–Strominger 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.^[17] Duncan and Frenkel produced additional evidence for this duality by usingRademacher sums to produce the McKay–Thompson series as (2 + 1)-dimensional gravity partition functions by a regularized sum over global torus-isogeny geometries.^[18] 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.

In 2010,Tohru Eguchi,Hirosi Ooguri, and Yuji Tachikawa observed that the elliptic genus of aK3 surface can be decomposed into characters of theN = (4,4)superconformal algebra, such that the multiplicities ofmassive states appear to be simple combinations of irreducible representations of theMathieu group M24.^[19] This suggests that there is asigma-modelconformal field theory with K3 target that carries M24 symmetry. However, by the Mukai–Kondo classification, there is nofaithful action of this group on any K3 surface bysymplectic automorphisms, and by work of Gaberdiel–Hohenegger–Volpato,^[20] there is no faithful action on any K3 sigma-model conformal field theory, so the appearance of an action on the underlyingHilbert space is still a mystery.

By analogy with McKay–Thompson series,Cheng suggested that both themultiplicity functions and the graded traces of nontrivial elements of M24 formmock modular forms. In 2012, Gannon proved that all but the first of the multiplicities are non-negativeintegral combinations of representations of M24, and Gaberdiel–Persson–Ronellenfitsch–Volpato computed all analogues of generalized moonshine functions,^[21] strongly suggesting that some analogue of a holomorphic conformal field theory lies behind Mathieu moonshine. Also in 2012, Cheng, Duncan, andHarvey amassed numerical evidence of anumbral moonshine phenomenon where families of mock modular forms appear to be attached toNiemeier lattices. The special case of theA²⁴
₁ lattice yields Mathieu moonshine, but in general the phenomenon does not yet have an interpretation in terms of geometry.

The term "monstrous moonshine" was coined by Conway, who, when told byJohn McKay in the late 1970s that the coefficient of${\displaystyle q}$ (namely 196884) was precisely one more than the degree of the smallest faithful complex representation of the monster group (namely 196883), replied that this was "moonshine" (in the sense of being a crazy or foolish idea).^[b] Thus, the term not only refers to the monster groupM; it also refers to the perceived lunacy of the intricate relationship betweenM and the theory of modular functions.

The monster group was investigated in the 1970s bymathematiciansJean-Pierre Serre,Andrew Ogg andJohn G. Thompson; they studied thequotient of thehyperbolic plane bysubgroups of SL₂(R), particularly, thenormalizer Γ₀(p)⁺ of theHecke congruence subgroup Γ₀(p) in SL(2,R). They found that theRiemann surface resulting from taking the quotient of thehyperbolic plane by Γ₀(p)⁺ hasgenus zero exactly forp = 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 theprime factors of the size ofM, he published a paper offering a bottle ofJack Daniel's whiskey to anyone who could explain this fact.^[22]These 15 primes are known as thesupersingular primes, not to be confused with the use of thesame phrase with a different meaning in algebraic number theory.

• Moonshine Bibliography at theWayback Machine (archived December 5, 2006)
• Mathematicians Chase Moonshine's Shadow

• Moonshine theory
• 1970s neologisms
• Group theory
• John Horton Conway
• Sporadic groups

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