← Modular Forms

Part 12 ended with Thompson's reading: the coefficients of J look like the dimensions of the graded pieces of one Monster representation. Conway and Norton turned that into a specification, and Frenkel, Lepowsky and Meurman (FLM) met it, announcing the moonshine module V^♮ in 1984 and giving the full construction in their 1988 book. Their tool was the vertex operator algebra (VOA), a framework from string theory and conformal field theory. Throughout, j = q⁻¹ + 744 + … is the classical invariant and J = j − 744 = q⁻¹ + 196 884q + … drops its constant term.

The specification

A graded space V = ⊕ V_n with finite-dimensional pieces has a graded dimension Σ dim V_n qⁿ. If a group acts on every piece, each element g has a graded trace:

For g = 1 the traces are dimensions. For other elements they are character values, so the series T_g for all g together fix how every grade splits into irreducibles.

The Conway–Norton conjecture (1979) asks for a graded Monster module V^♮ = ⊕_{n ≥ −1} V_n^♮ with graded dimension J, such that every T_g is the Hauptmodul of a genus-zero group: a single modular function that generates every function invariant under that group, as j does for SL₂(ℤ). One exists exactly when the compactified quotient of ℍ by the group has genus zero. Conway and Norton named a candidate for each of the 194 conjugacy classes; algebraically conjugate classes such as 23A and 23B share one, so there are 171 distinct functions.

The first grades fix the shape. Grade −1 is one trivial state. Grade 0 is empty: in a VOA it would be a Lie algebra, its exponential would be a continuous group of symmetries, and a finite automorphism group needs it gone. Grade 1 has dimension 196 884, splits as 1 ⊕ 196 883 (Part 12 assembles this grade and the next two from irreducibles), and carries the commutative non-associative product Griess used in 1982.

Vertex operator algebras

In a VOA each state v acts on every other through its vertex operator, a formal series of linear maps v_(n):

A VOA is a space V = ⊕_n≥0 V_n graded by weight, with a vacuum |0⟩ in V₀ and a conformal vector ω in V₂, under four axioms: the vacuum acts as the identity; a translation operator differentiates every Y(v, z); any two vertex operators commute up to a pole of finite order, (z − w)^N[Y(u, z), Y(v, w)] = 0; and the modes of Y(ω, z) = Σ L_n z⁻ⁿ⁻² form a Virasoro algebra with central charge c:

A theory with c = 24 has graded dimension Σ dim V_k q^k−1, so the vacuum gives q⁻¹, the empty grade 0 of J is the weight-1 space, and the Griess algebra is the weight-2 space.

Free bosons and lattices

The simplest VOA is a free boson, with c = 1: polynomials in α₋₁, α₋₂, …, where α_−n has weight n, so the states of weight n are counted by the partitions of n:

d bosons give c = d. Adding a momentum state e^α of weight |α|²/2 for each vector α of an even lattice Λ ⊂ ℝ^d gives the lattice VOA V_Λ, a string on the torus ℝ^d/Λ, with graded dimension Θ_Λ/η^d. For the 24-dimensional Leech lattice that is j − 720:

The weight-1 space of V_Λ holds the oscillators αⁱ₋₁ and one state e^α for each root, a vector of norm 2. Together they form a Lie algebra, a continuous symmetry V^♮ must not have. The even unimodular lattices in 24 dimensions are where to look: Niemeier classified them in 1973, and there are 24. Each theta series is a modular form of weight 12 with constant term 1, and that space is spanned by E₄³ and Δ, so the number of roots N₂ fixes the rest:

Every root costs 24 vectors of norm 4, and every lattice VOA gets 24 + N₂ states at weight 1 but exactly 196 884 at weight 2. Only the Leech lattice Λ₂₄ has no roots, and even its VOA keeps the 24 oscillators, so V^♮ has to be built from V_Λ24 rather than be it.

The FLM orbifold

The involution θ: x ↦ −x of the Leech lattice lifts to V_Λ24, acting as −1 on each oscillator and sending e^α to ±e^−α. Its fixed subspace V_Λ24⁺ loses the 24 oscillators at weight 1, which are odd, but also about half of everything above. Orbifold theory supplies the replacement: a twisted sector V_Λ24^T of strings that close only up to θ, with 2¹² ground states at weight 3/2. Keeping the θ-fixed part of each:

At weight 1 the orbifold drops the 24 oscillators and adds nothing, since the twisted sector starts at weight 3/2: that is the empty grade 0 of J. At weight 2 it keeps 300 symmetric pairs of oscillators and half of the 196 560 shortest vectors, 98 580 states, and adds 2¹²·24 = 98 304. From weight 2 on the dropped and added counts are equal, because V_Λ24 already had the right dimensions there: its graded dimension j − 720 differs from J only in the constant term.

The Monster as automorphism group

FLM's main theorem is Aut(V^♮) = 𝕄. The weight-2 space inherits a product from the vertex operator, the v₍₁₎w coefficient, which makes it the 196 884-dimensional Griess algebra. Griess built the Monster as a group of automorphisms of this algebra (published 1982), and Tits proved in 1983–84 that it is the full automorphism group. V^♮ is generated as a VOA by its weight-2 space, so an automorphism is fixed by what it does there and Aut(V^♮) embeds in 𝕄; FLM show every Monster element extends, so the embedding is onto.

V_Λ24⁺ has automorphism group 2²⁴·Co₁, where Co₁ is the automorphism group of the Leech lattice modulo ±1. The symmetries of V^♮ that keep the two sectors apart form 2¹⁺²⁴·Co₁, the centraliser of an involution of class 2B in 𝕄; the extension does not split, so Co₁ appears there only as a quotient. One more automorphism, mixing the sectors, generates the rest of 𝕄.

What remained

FLM could compute the graded traces T_g for elements of 2¹⁺²⁴·Co₁, but not for all 194 classes. Proving that every T_g is the predicted Hauptmodul took a Lie algebra whose denominator identity is the product formula for j. That is Borcherds' work, the subject of Part 14.