Monstrous moonshine

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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.


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,

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 \\ 

(Since there are many linear relations between the rn such as r_1 - r_3 + r_4 + r_5 - r_6 = 0, then 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.



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.


The Monster module[edit]

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


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[edit]


  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.
  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.
  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.


Generalized Moonshine[edit]


In 1980, Larissa Queen and others subsequently found that one can in fact construct the expansions of many Hauptmodul (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:

  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.


Conjectured relationship with quantum gravity[edit]




Mathieu Moonshine[edit]



Why "monstrous moonshine"?[edit]

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.


See also[edit]


  3. ^ World Wide Words: Moonshine


External links[edit]