Monstrous moonshine

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) (sequence A000521 in OEIS),

j(\tau) = \frac{1}{{q}} + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + 20245856256{q}^4 + \cdots

with {q} = e^{2\pi i\tau} 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 (sequence A001379 in OEIS) with small non-negative coefficients. Let r_n = 1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, ... then,

\begin{align} 1 & = r_1 \\ 196884 & = r_1 + r_2 \\ 21493760 & = r_1 + r_2 + r_3 \\ 864299970 & = 2r_1 + 2r_2 + r_3 + r_4 \\ 20245856256& = 3r_1 + 3r_2 + r_3 + 2r_4+ r_5\\ 333202640600 & =5r_1 +5r_2+2r_3+3r_4+2r_5+r_7 \\ \end{align}

(Since there can be several linear relations between the rn such as r_1 - r_3 + r_4 + r_5 - r_6 = 0, 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 Tg, and found that all of them appeared to be the expansions of Hauptmoduln. In other words, if Gg is the subgroup of SL2(R) which fixes Tg, then the quotient of the upper half of the complex plane by Gg is a sphere with a finite number of points removed, and furthermore, Tg 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 Tg 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 V^\natural, 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.

The monster module

The Frenkel–Lepowsky–Meurman construction uses two main tools:

  1. The construction of a lattice vertex operator algebra VL for an even lattice L of rank n. In physical terms, this is the chiral algebra for a bosonic string compactified on a torus Rn/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. Attached to the –1 involution of the Leech lattice, there is an involution h of VL, and an irreducible h-twisted VLmodule, which inherits an involution lifting h. To get the Moonshine Module, one takes the fixed point subspace of h in the direct sum of VL and its twisted module.

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 (Frenkel, Lepowsky & Meurman (1988)).

Borcherds' proof

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 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.
  2. A Lie algebra \mathfrak{m}, called the monster Lie algebra, is constructed from V using a quantization functor. It is a generalized Kac–Moody Lie algebra with a monster action by automorphisms. Using the Goddard–Thorn "no-ghost" theorem from string theory, the root multiplicities are found to be coefficients of j.
  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 that Hecke operators applied to j yield polynomials in j.
  4. By comparing root multiplicities, one finds that the two Lie algebras are isomorphic, and in particular, the Weyl denominator formula for \mathfrak{m} is precisely the Koike–Norton–Zagier identity.
  5. Using Lie algebra homology and Adams operations, a twisted denominator identity is given for each element. These identities are related to the McKay–Thompson series Tg in much the same way that the Koike–Norton–Zagier identity is related to j.
  6. The twisted denominator identities imply recursion relations on the coefficients of Tg. 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.

Thus, the proof is completed (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."[1]

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.[2]

In 1980, Larissa Queen and others subsequently found that one can in fact construct the expansions of many Hauptmoduln (McKay-Thompson series Tg) from simple combinations of dimensions of sporadic groups. 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:

  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.
  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 \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}), f(g,h,\frac{a\tau+b}{c\tau+d}) is proportional to f(g^a h^c, g^b h^d, \tau).
  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. To date, this conjecture is still open.

Like the Conway–Norton conjecture, Generalized Moonshine also has an interpretation in physics, proposed by Dixon–Ginsparg–Harvey in 1988 (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 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.

Conjectured relationship with quantum gravity

In 2007, E. Witten suggested that AdS/CFT correspondence yields a duality between pure quantum gravity in (2+1)-dimensional 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.

Under Witten's proposal (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 Bekenstein-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 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 (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 (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 (Duncan & Frenkel (2009)) produced additional evidence for this duality by using Rademacher sums 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.

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 N=(4,4) 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 A124 lattice yields Mathieu Moonshine, but in general the phenomenon does not yet have an interpretation in terms of geometry.

Origin of the term

The term "monstrous moonshine" was coined by Conway, who, when told by John McKay in the late 1970s that the coefficient of {q} (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 "moonshine" (in the sense of being a crazy or foolish idea).[3] 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 functions.

The monster group was investigated in the 1970s by mathematicians Jean-Pierre Serre, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of Γ0(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has 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 factors of the size of M, he published a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact (Ogg (1974)).

Notes

  1. ↑ Roberts, Siobhan (2009), King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, Bloomsbury Publishing USA, p. 361, ISBN 9780802718327.
  2. ↑ Conway, J. and Norton, S. "Monstrous Moonshine", Table 2a, p.330, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.103.3704&rep=rep1&type=pdf
  3. ↑ World Wide Words: Moonshine

References

External links

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