In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular the j function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979. The term "moonshine" was used by Conway in the sense of "foolish or crazy ideas", reflecting the seemingly absurd nature of the connection; as number theorist Don Zagier later put it, "they called it moonshine because it appeared so far-fetched."

The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.

History

The Jack Daniel's Problem

In the mid-1970s, mathematicians Jean-Pierre Serre, Andrew Ogg and John G. Thompson studied the quotient of the hyperbolic plane by subgroups of SL<sub>2</sub>(R), particularly, the normalizer Γ<sub>0</sub>(p)<sup>+</sup> of the Hecke congruence subgroup 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 zero exactly for p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg attended a lecture by Jacques Tits in which the conjectural order of the monster group was presented, he noticed that these were precisely the prime factors of the group's size. He published a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this coincidence. This observation is widely considered the earliest hint of monstrous moonshine, and the challenge of explaining it became known as "The Jack Daniel's Problem". These 15 primes are now known as the supersingular primes, not to be confused with the use of the same phrase with a different meaning in algebraic number theory.

Although Richard Borcherds's 1992 proof of the monstrous moonshine conjecture established a deep connection between the monster group and modular functions, it provides a route from the monster to the genus-zero property but not in the reverse direction, leaving Ogg's original question not fully resolved. , the bottle of Jack Daniel's remains unclaimed.

McKay's observation and the Conway–Norton conjecture

In 1978, John McKay found that the first few terms in the Fourier expansion of the normalized J-invariant could be expressed in terms of linear combinations of the dimensions of the irreducible representations <math>r_n</math> of the monster group M with small non-negative coefficients. A graded representation whose graded dimension is J, called the moonshine module, was explicitly constructed by Igor Frenkel, James Lepowsky, and Arne 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 of M, partially settling the Conway–Norton conjecture.

Borcherds proved the Conway–Norton conjecture for the Moonshine Module in 1992.

The moonshine module

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

  1. The construction of a lattice vertex operator algebra V<sub>L</sub> for an even lattice L of rank n. In physical terms, this is the chiral algebra for a bosonic string 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 infinitely many generators). For the case in question, one sets L to be the Leech lattice, which has rank 24.
  2. The orbifold construction. In physical terms, this describes a bosonic string propagating on a quotient orbifold. The construction of Frenkel–Lepowsky–Meurman was the first time orbifolds appeared in conformal field theory.

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 of gl<sub>2</sub> and two free Lie algebras. 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.

Generalized moonshine

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 particular, she decomposed the coefficients of McKay-Thompson series into representations of subquotients of the Monster in the following cases:

  • T<sub>2B</sub> and T<sub>4A</sub> into representations of the Conway group Co<sub>0</sub>
  • T<sub>3B</sub> and T<sub>6B</sub> into representations of the Suzuki group 3.2.Suz
  • T<sub>3C</sub> into representations of the Thompson group Th = F<sub>3</sub>
  • T<sub>5A</sub> into representations of the Harada–Norton group HN = F<sub>5</sub>
  • T<sub>5B</sub> and T<sub>10D</sub> into representations of the Hall–Janko group 2.HJ
  • T<sub>7A</sub> into representations of the Held group He = F<sub>7</sub>
  • T<sub>7B</sub> and T<sub>14C</sub> into representations of 2.A<sub>7</sub>
  • T<sub>11A</sub> into representations of the Mathieu group 2.M<sub>12</sub>

Queen found that the traces of non-identity elements also yielded q-expansions of Hauptmoduln, some of which were not McKay–Thompson series from the Monster. 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:

  1. Each V(g) is a graded projective representation of the centralizer of g in M.
  2. Each f(g, h, τ) is either a constant function, or a Hauptmodul.
  3. Each f(g, h, τ) is invariant under simultaneous conjugation of g and h in M, up to a scalar ambiguity.
  4. 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.
  5. For any <math>(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}) \in \operatorname{SL}_2(\mathbf{Z})</math>, <math>f(g, h, \tfrac{a\tau + b}{c\tau + d})</math> is proportional to <math>f(g^a h^c, g^b h^d, \tau)</math>.
  6. f(g, h, τ) is proportional to J if and only if g = h = 1.

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

Like the Conway–Norton conjecture, Generalized Moonshine also has an interpretation in physics, proposed by Dixon, Ginsparg, and Harvey in 1988. 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 one partition 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 by monodromy along a basis of 1-cycles, i.e., a pair of commuting elements.

Modular moonshine

In the early 1990s, the group theorist A. J. E. Ryba discovered remarkable similarities between parts of the character table of the monster, and Brauer characters of certain subgroups. In particular, for an element g of prime order p in the monster, many irreducible characters of an element of order kp whose k th power is g are simple combinations of Brauer characters for an element of order k in the centralizer of g. This was numerical evidence for a phenomenon similar to monstrous moonshine, but for representations in positive characteristic. In particular, Ryba conjectured in 1994 that for each prime factor p in the order of the monster, there exists a graded vertex algebra over the finite field F<sub>p</sub> with an action of the centralizer of an order p element g, such that the graded Brauer character of any p-regular automorphism h is equal to the McKay-Thompson series for gh.

In 1996, Borcherds and Ryba reinterpreted the conjecture as a statement about Tate cohomology of a self-dual integral form of <math>V^\natural</math>. This integral form was not known to exist, but they constructed a self-dual form over Z[1/2], which allowed them to work with odd primes p. The Tate cohomology for an element of prime order naturally has the structure of a super vertex algebra over F<sub>p</sub>, 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. In 1998, Borcherds showed that vanishing holds for the remaining odd primes, using a combination of Hodge theory and an integral refinement of the no-ghost theorem.

The case of order 2 requires the existence of a form of <math>V^\natural</math> 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.

Mathieu moonshine

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 superconformal algebra, such that the multiplicities of massive states 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 symplectic automorphisms, 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.

By analogy with McKay–Thompson series, Cheng suggested that both the multiplicity functions and the graded traces of nontrivial elements of M24 form mock modular forms. In 2012, Gannon proved that all but the first of the multiplicities are non-negative integral combinations 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 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 lattice yields Mathieu moonshine, but in general the phenomenon does not yet have an interpretation in terms of geometry.

Conjectured relationship with quantum gravity

In 2007, E. Witten suggested that AdS/CFT correspondence yields a duality between pure quantum gravity in (2&nbsp;+&nbsp;1)-dimensional anti de Sitter space and extremal holomorphic CFTs. Pure gravity in 2&nbsp;+&nbsp;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.

Under Witten's proposal, 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. 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. Duncan and Frenkel produced additional evidence for this duality by using Rademacher sums to produce the McKay–Thompson series as (2&nbsp;+&nbsp;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.

Notes

References

Sources

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  • (Provides introductory reviews to applications in physics).
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  • (The first book about the Monster Group written in Japanese).
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  • (Concise introduction for the lay reader).
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  • Mathematicians Chase Moonshine's Shadow