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) ā j(q).
Part 13 left the Conway–Norton conjecture open: FLM could compute graded traces only on one subgroup of the Monster. Richard Borcherds proved it for all 194 classes in 1992 by building a Lie algebra from Vā®, reading its denominator identity as the product formula for j, and twisting that identity by Monster elements into recursions so rigid that four coefficients of each graded trace fix the rest. Throughout, j(p) − j(q) = J(p) − J(q), and c(n) is the coefficient of qn in J, with c(0) = 0.
A Kac–Moody algebra is built from a Cartan matrix with aii = 2, so every simple root has positive norm and multiplicity 1. Finite-dimensional simple Lie algebras have only such real roots; affine algebras add the multiples of a null vector δ, with multiplicity at most the rank. Neither can carry multiplicities that grow like the j-coefficients, as e4π√n.
Borcherds also allowed aii ≤ 0:
A simple root with aii ≤ 0 is imaginary: it has no reflection and may have multiplicity above 1. These generalised Kac–Moody (GKM) algebras, or Borcherds algebras, keep root spaces, a triangular decomposition and a Weyl group W generated by the real reflections, and Weyl's denominator identity gains a correction for the imaginary simple roots:
Here ρ is the Weyl vector and S runs over sets of pairwise orthogonal imaginary simple roots, with sum σ(S). With no imaginary roots the inner sum is 1 and this is Weyl's identity.
Goddard and Thorn's no-ghost theorem (1972) says that in 26 dimensions the bosonic string's physical states match those built from its 24 transverse directions. Borcherds tensored Vā® (c = 24) with the vertex algebra of the Lorentzian lattice II1,1 to reach c = 26, and took the physical states modulo the null vectors of their bilinear form. The result is a Lie algebra šŖ graded by II1,1 = ℤ2, with (m, n)·(m′, n′) = −(mn′ + m′n). By the no-ghost theorem its piece of degree (m, n) ≠ (0, 0) is Vmnā® as a Monster module, of dimension c(mn); the (0, 0) piece is a two-dimensional Cartan subalgebra.
II1,1 near the origin. Circles are roots of šŖ, growing with the number of digits of c(mn), computed from E43/Δ. The axes are the null cone. Simple roots are ringed: the real root (1, −1) and the column m = 1. Roots on the selected hyperbola share a multiplicity.
So (m, n) is a root exactly when mn = −1 or mn ≥ 1, and its norm −2mn fixes its multiplicity. The simple roots are (1, −1), the only real one, with norm 2 and multiplicity 1, and (1, n) for n ≥ 1, imaginary, with multiplicity c(n). Reflection in (1, −1) swaps m and n, so W has two elements. No two imaginary simple roots are orthogonal, since (1, n)·(1, n′) = −(n + n′), so each S is empty or a single root. Writing p and q for e−(1,0) and e−(0,1), eρ is p−1, and the sum side is already the j-function:
The product side runs over the positive roots, (1, −1) and every (m, n) with m, n ≥ 1. Equating the two sides gives
the product formula that Koike, Norton and Zagier each found and proved in the 1980s. Borcherds used it in reverse, to identify the simple roots of šŖ and their multiplicities.
The left side multiplied out exactly, with exponents c(mn). Cell (a, b) is the coefficient of paqb. Row a = 0 is −J(q), column b = 0 is J(p), and every mixed coefficient is 0, though its cancelling halves reach 2.3 × 1016 at the largest setting.
The Monster acts on šŖ because šŖ is built from Vā®. The denominator identity comes from comparing exterior powers of the positive part of šŖ with its Lie algebra homology; taking the trace of g on both sides instead of the dimension gives a twisted denominator formula:
It ties Tg to the series Tgk of the powers of g, and comparing coefficients gives replication formulas. For g = 1, comparing coefficients in J(τ)2 − 2c(1) = J(2τ) + J(τ/2) + J((τ+1)/2) gives each c(2n), n ≥ 2, from coefficients with smaller index; at n = 1 it reads c(2) = c(2).
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). The arcs run from the coefficients it uses (purple) to the target c(2n) (orange); the readout compares the result with E43/Δ expanded directly.
The same identity at order 4 reaches the odd indices except 1, 3 and 5; for J these recursions go back to Mahler (1974). For other g they also use the series of the powers of g, and they leave ag(1), ag(2), ag(3) and ag(5) free.
A graded trace two ways, signed log scale: the Hauptmodul's expansion (rings), and what the order-2 and order-4 formulas build from four seeds (squares) and the series of g2 and g4 (dots). For 2B they agree at all 24 coefficients. Expansions checked against OEIS.
The two formulas accept any seeds. The order-3 identity, which the cascade never uses, holds for the true seeds of all five classes. Of the 120 ways to change one seed of one class by 1 to 3, 88 run through and break that identity at q3; the other 32, odd changes for 2A and 2B, give a half-integer coefficient, which no trace can have.
Norton had conjectured, and Koike proved, that Conway and Norton's candidate functions satisfy these replication formulas. Borcherds showed the graded traces of Vā® satisfy them too, since they come from the twisted identities. So it was enough to match ag(1), ag(2), ag(3) and ag(5) for each class, the traces of g on V1ā®, V2ā®, V3ā® and V5ā®. Only the first seven Monster irreducibles fit in those spaces, so Borcherds fixed their decompositions from traces at seven elements of 21+24·Co1, where FLM's trace formula applies, and the check holds for every class. The citation for his 1998 Fields Medal names the proof of the Conway–Norton conjecture.
In 2010 Eguchi, Ooguri and Tachikawa noticed that the elliptic genus of a K3 surface, expanded in characters of the N = 4 superconformal algebra, has multiplicities that are sums of dimensions of representations of the Mathieu group M24, of order 244 823 040. Cheng, Gaberdiel–Hohenegger–Volpato and Eguchi–Hikami found candidate twined genera for every element, and Gannon proved in 2012 (published 2016) that they come from genuine M24 representations.
Cheng, Duncan and Harvey extended this to six groups in 2012 and in 2013 to umbral moonshine, one instance for each of the 23 Niemeier lattices with roots (Part 13), pairing a finite group with vector-valued mock modular forms; Duncan, Griffin and Ono proved the 22 cases beyond M24 in 2015. Leech, with no roots, has no instance: it is the lattice Vā® is built from.
Moonshine has since been found for the O’Nan group (Duncan, Mertens and Ono, with forms of weight 3/2) and the Thompson group (proposed by Harvey and Rayhaun, proved by Griffin and Mertens, weight 1/2). The E8 and Leech lattices are the subject of the companion Exceptional Atlas series.