Part 14 of 14. Borcherds proved the moonshine conjecture in 1992 with a Lie algebra whose denominator identity is the product formula for j(p) − j(q).
Part 13 ended with V♮ built but the Conway–Norton conjecture unproved: FLM could compute graded traces only for elements of one subgroup of the Monster. To prove monstrous moonshine, Richard Borcherds had to show that the graded traces of the moonshine module are the Hauptmoduls Conway and Norton predicted, for all 194 conjugacy classes. He introduced generalised Kac–Moody algebras and used the no-ghost theorem from string theory to extract a Monster Lie algebra from V♮.
A note on notation: j(τ) = q−1 + 744 + … is the classical modular invariant, while J(τ) = j(τ) − 744 = q−1 + 0 + 196884q + … is the moonshine-normalised graded dimension. Borcherds' product formula has j(p) − j(q) on the right, which equals J(p) − J(q), so the constant term drops out. Root multiplicities are the J coefficients, so c(0) = 0 throughout this page.
The machinery yields replication formulas, rigid enough that the coefficients at n = 1, 2, 3 and 5 of a McKay–Thompson series, together with the series for the powers of g, fix every other coefficient. The figure below lays out the steps.
The proof pipeline, from V♮ to a Lie algebra, its denominator identity, and the twisted identities that force the Conway–Norton trace functions.
Finite-dimensional simple Lie algebras have all-positive root norms; affine Kac–Moody adds null directions, but its root multiplicities stay bounded (at most the rank). Neither matches the j-coefficients' e4π√n growth.
The constraint is in the Cartan matrix: a Kac–Moody algebra has aii = 2, so every simple root has positive norm and multiplicity 1.
Borcherds kept aii = 2 and also allowed aii ≤ 0. Simple roots with aii = 2 behave classically, with an sl₂ and a Weyl reflection. Simple roots with aii ≤ 0 are imaginary: they have non-positive norm, no reflection, and possibly multiplicity greater than 1.
These are generalised Kac–Moody (GKM) algebras, or Borcherds algebras. Root spaces, the triangular decomposition and the Weyl group all survive, and imaginary simple roots with large multiplicities allow the growth of the j-coefficients.
A real root lies in the Weyl orbit of a positive-norm simple root and has multiplicity 1. Every other root is imaginary, with zero or negative norm and possibly multiplicity greater than 1.
Finite Lie algebras have only real roots. Affine algebras have imaginary roots only as multiples of δ. GKM algebras can have many imaginary roots with large multiplicities.
The root system of A₂ (= sl3), whose six roots are all real, and a schematic GKM root system in a Lorentzian plane. Real roots (positive norm, outside the dashed light cone) are filled circles; imaginary roots (inside it) are open amber circles, with illustrative multiplicities.
Weyl's denominator identity equates a product over positive roots with a signed sum over the Weyl group:
Every GKM algebra has a version with a correction for imaginary simple roots:
W is the Weyl group (real-root reflections only), ρ is the Weyl vector. The correction runs over subsets S of pairwise-orthogonal imaginary simple roots; (−1)|S| alternates the sum and σ(S) = Σα∈S α.
Borcherds built a GKM algebra from V♮ whose denominator formula is a j-function identity, using a result from 1970s bosonic string theory. Goddard and Thorn (1972) proved that in 26 dimensions the physical states of the bosonic string have positive norm and are isomorphic to the states built from the 24 transverse directions. Above 26 dimensions negative-norm states appear.
V♮ at central charge 24 embeds into the Fock space of a 26-dimensional bosonic string, 24 transverse directions plus 2 lightcone ones. The physical states of that string, taken modulo the null vectors of their bilinear form, form a ℤ²-graded Lie algebra, and the no-ghost theorem identifies each graded piece with a piece of V♮.
The Monster Lie algebra 𝔪 is ℤ²-graded:
For (m, n) ≠ (0, 0), whatever the signs, 𝔪(m,n) is isomorphic as a Monster representation to Vmn♮. The (0, 0) piece is a two-dimensional Cartan subalgebra coming from the two lightcone directions. So the (m, n) piece has dimension c(mn), the coefficient of qmn in J. Since c(0) = 0 and c(k) = 0 for k < −1, the pieces with mn = 0 or mn ≤ −2 are empty. Because the null vectors have been divided out, 𝔪 carries a non-degenerate invariant bilinear form.
The construction tensors V♮ with the vertex algebra of the two-dimensional Lorentzian lattice II1,1, giving central charge 24 + 2 = 26, the value the no-ghost theorem needs. V♮ has c = 24 because Λ24 is 24-dimensional.
The root lattice of 𝔪 is II1,1 = ℤ2 with the form (m, n)·(m′, n′) = −(mn′ + m′n), so (m, n) has norm −2mn. The simple roots are (1, n) for n = −1 and n ≥ 1, with multiplicity c(n); (1, n) has norm −2n. The root (1, −1) has norm 2 and multiplicity c(−1) = 1, and it is the only real simple root. The simple roots (1, n) for n ≥ 1 have norm −2n and are imaginary, with multiplicities c(1) = 196 884, c(2) = 21 493 760, c(3) = 864 299 970, growing exponentially as the GKM framework allows.
Applying the GKM denominator formula to 𝔪 gives:
This is the Koike–Norton–Zagier product formula, which Koike, Norton and Zagier proved independently in the 1980s. Borcherds read it as a denominator formula, and comparing the two is how he identified the simple roots of 𝔪 and their multiplicities. The left side encodes the root system of 𝔪, the root multiplicities are the graded dimensions of V♮, and the right side is j(p) − j(q), so the identity relates the Lie algebra, the vertex algebra and the modular function.
The left side of the identity, multiplied out exactly in the page for p- and q-degrees below the slider value, using exponents c(mn) from E43/Δ. Row a, column b holds the coefficient of paqb. The a = 0 row reproduces −J(q) and the b = 0 column reproduces J(p); every mixed coefficient is 0, although the two halves that cancel there reach 2.3 × 1016 at the largest setting. Click a cell to see its two halves.
The Koike–Norton–Zagier identity is the case of the identity element: it matches the graded dimension of V♮ with the j-function. The Conway–Norton conjecture asks that the graded trace of every Monster element be a specific Hauptmodul Tg(τ).
The Monster acts on 𝔪 because 𝔪 was built from V♮. For each g, Borcherds replaced dimensions of the graded pieces by traces of g on them. The denominator formula comes from comparing the exterior powers of the positive part of 𝔪 with its Lie algebra homology (an Euler–Poincaré principle); taking the trace of g on both sides instead of the dimension produces a “twisted denominator formula”:
Here Tg is the McKay–Thompson series of g, and the exponents involve traces of powers of g on Vn♮. The identity relates the unknown Tg to itself and to the series Tgk for powers of g.
From the twisted identities Borcherds extracted replication formulas: recurrences that express a Fourier coefficient of Tg through coefficients at smaller indices. Write Tg(τ) = ∑ ag(n) qn. The formulas give ag(n) for every n other than 1, 2, 3 and 5 as a polynomial in coefficients at smaller indices, of Tg itself and of the series Tgd for powers of g.
The simplest case is g = 1, where every Tgd is J. Comparing coefficients in J(τ)2 − 2c(1) = J(2τ) + J(τ/2) + J((τ+1)/2) gives every even coefficient c(2n), n ≥ 2, from coefficients with smaller index. The corresponding identity at order 4 reaches the odd indices, apart from 1, 3 and 5; for J these recursions go back to Mahler (1974). (At n = 1 the formula below reads c(2) = c(2) and says nothing.)
So once the coefficients at n = 1, 2, 3 and 5 are known, together with the series for the powers of g, every other coefficient is determined. Two series that satisfy the twisted identities and agree at those four indices agree everywhere.
The degree-2 replication formula for J, c(2n) = c(n+1) + ½Σi=1n−1 c(i)c(n−i) − ½[n even]c(n/2). Slide n to pick the target c(2n) (orange); the arcs come from the coefficients the formula uses (purple), all at smaller indices. The page evaluates the formula and compares it with the coefficient from expanding E43/Δ directly.
Norton had conjectured, and Koike proved, that Conway and Norton's Hauptmodul candidates (171 distinct functions for the 194 classes) satisfy the same replication formulas. Borcherds showed that the graded traces of V♮ satisfy them too, since they come from the twisted denominator identities. What remains is to check the initial coefficients.
For each of the 194 classes, ag(1), ag(2), ag(3) and ag(5) must equal the corresponding coefficients of the conjectured Hauptmodul. These are traces of g on V1♮, V2♮, V3♮ and V5♮. Only the first seven Monster irreducibles are small enough to occur in V1♮ through V5♮, so Borcherds fixed how those spaces decompose by evaluating traces at seven well-chosen elements of 21+24·Co1, where FLM's trace formula applies. The check is finite, and it holds for every class. The citation for Borcherds' 1998 Fields Medal names the proof of the Conway–Norton conjecture.
Borcherds’ proof settled the Conway–Norton conjecture. What remained open was why moonshine happens, and whether other groups have their own.
In 2010, Eguchi, Ooguri, and Tachikawa noticed that when the elliptic genus of a K3 surface is expanded in characters of the N = 4 superconformal algebra, the multiplicities are sums of dimensions of representations of the Mathieu group M24, a sporadic simple group of order 244 823 040 that acts on the Steiner system S(5, 8, 24). The coefficients of a weak Jacobi form encode M24 representations, just as the coefficients of J encode Monster representations. Cheng, Gaberdiel–Hohenegger–Volpato and Eguchi–Hikami then found a candidate twined genus for every element of M24. Gannon proved in 2012 (published 2016) that the resulting class functions are characters of genuine M24 representations, which proves the conjecture.
Cheng, Duncan, and Harvey generalised Mathieu Moonshine, first to six groups in 2012 and then in 2013 to a family of 23 “umbral moonshine” instances, one for each of the 23 Niemeier lattices with roots (the Niemeier lattices are the 24 even unimodular positive-definite lattices of rank 24). Each instance pairs a finite group with vector-valued mock modular forms. Gannon's theorem covers the M24 case, and Duncan, Griffin and Ono proved the remaining 22 in 2015. The 24th Niemeier lattice, Leech, has no roots and no umbral instance; it is the lattice the moonshine module itself is built from.
Duncan, Mertens, and Ono established moonshine for the O’Nan group, via weight-3/2 modular forms related to arithmetic invariants of elliptic curves. Harvey and Rayhaun proposed moonshine for the Thompson group, via weakly holomorphic modular forms of weight 1/2, and Griffin and Mertens proved it. The modular objects in both cases are not Hauptmoduls, but the pattern is the same: a sporadic group acts on a graded space whose graded traces are distinguished modular objects.
The lattices E8 and Leech are the subject of the companion Exceptional Atlas series.