Leech Pair: Lattice and VOA Perspectives

• Leech pair refers to two interrelated frameworks: a lattice-theoretic pair (G, S) with S being an even, positive-definite rootless lattice, and a VOA-related pair (τ, 𝛽̃) linked to deep holes in the Leech lattice.
• In the lattice setting, the pair (G, S) must satisfy a numerical constraint (rank(S) + l(A_S) ≤ 24) ensuring a primitive embedding into the Leech lattice with G acting trivially on the orthogonal complement.
• The VOA approach encodes orbifold data via coinvariant lattices and deep holes, facilitating a combinatorial classification of holomorphic VOAs of central charge 24 through Conway-group and Niemeier-lattice methods.

Searching arXiv for papers on "Leech pair" and closely related lattice/VOA context. A Leech pair is a term used in two closely related but non-identical lattice-theoretic senses centered on the Leech lattice $\Lambda$. In "A Lemma on Leech-like Lattices" (Zheng, 14 Jul 2025), a Leech pair is a pair $(G,S)$ consisting of a finite group $G$ acting on an even positive-definite rootless lattice $S$ with trivial fixed sublattice and trivial induced action on the discriminant group. In the Lam–Miyamoto framework summarized in (Lam et al., 2022), a Leech pair is a pair $(\tau,\tilde\beta)$ attached to generalized deep holes of the Leech lattice vertex operator algebra $V_\Lambda$. Both usages organize data that can be transferred to the Leech lattice and the Conway group $Co_0=O(\Lambda)$, but they arise in different classification problems: the former in the study of symplectic automorphisms of hyperkähler-type geometries, the latter in the classification of holomorphic vertex operator algebras of central charge $24$.

1. The lattice-theoretic definition $(G,S)$

Let $S$ be an even, positive-definite lattice of rank $(G,S)$0. Its dual lattice is

$(G,S)$1

and its discriminant group is

$(G,S)$2

an abelian group equipped with the natural $(G,S)$3-valued quadratic form induced by $(G,S)$4 on $(G,S)$5. The quantity $(G,S)$6 is the minimal number of generators of $(G,S)$7, referred to as its length (Zheng, 14 Jul 2025).

A pair $(G,S)$8 is called a Leech pair when the following conditions hold. First, $(G,S)$9 is even, positive definite, and rootless, meaning that there is no $G$0 with $G$1. Second, $G$2 acts faithfully on $G$3 and has no nonzero invariants:

$G$4

Third, every $G$5 induces the identity on $G$6, equivalently

$G$7

The same condition can be stated by saying that $G$8 fixes $G$9 pointwise modulo $S$0 and leaves no nonzero vector of $S$1 invariant (Zheng, 14 Jul 2025).

This definition isolates lattices that are simultaneously rootless and rigid under the given finite-group action. The absence of roots aligns the structure with the Leech lattice, while the triviality of the induced action on $S$2 is the hypothesis that later permits extension of the group action across a primitive embedding.

2. The embedding lemma and the numerical bound

Let $S$3 denote the Leech lattice, the unique even, positive-definite, unimodular, rootless lattice of rank $S$4. The central structural statement is the corrected Gaberdiel–Hohenegger–Volpato primitive embedding lemma: if $S$5 is a Leech pair and

$S$6

then there exists a primitive, $S$7-equivariant embedding

$S$8

such that $S$9 acts trivially on the orthogonal complement $(\tau,\tilde\beta)$0 in $(\tau,\tilde\beta)$1 (Zheng, 14 Jul 2025).

In the formulation given in (Zheng, 14 Jul 2025), the conclusion is stronger than mere existence of an embedding. The embedding is required to be primitive, the $(\tau,\tilde\beta)$2-action must extend from $(\tau,\tilde\beta)$3 to $(\tau,\tilde\beta)$4, and the extension is constrained so that

$(\tau,\tilde\beta)$5

This makes the lemma a transfer principle from an abstract rootless lattice with controlled discriminant action into the universal rank-$(\tau,\tilde\beta)$6 rootless unimodular setting.

The role of the bound $(\tau,\tilde\beta)$7 is explicit. It is used to invoke Nikulin’s theorem on primitive embeddings into the even unimodular lattice of signature $(\tau,\tilde\beta)$8. Without that numerical constraint, one cannot guarantee the existence of a primitive embedding of $(\tau,\tilde\beta)$9.

3. Corrected proof and the counterexample to the original argument

The corrected proof has two principal stages (Zheng, 14 Jul 2025). The first is an embedding into the even unimodular lattice

$V_\Lambda$0

By Nikulin’s theorem on primitive embeddings, the bound

$V_\Lambda$1

guarantees a primitive orthogonal embedding

$V_\Lambda$2

where $V_\Lambda$3 is the rank-one lattice of norm $V_\Lambda$4.

The second stage passes from $V_\Lambda$5 to the Leech lattice through a Conway chamber argument. Writing $V_\Lambda$6, one chooses the image $V_\Lambda$7 of the $V_\Lambda$8-generator and a component

$V_\Lambda$9

of $Co_0=O(\Lambda)$0, the Lobachevsky cone. One then studies

$Co_0=O(\Lambda)$1

which has $Co_0=O(\Lambda)$2 and meets the reflection hyperplane

$Co_0=O(\Lambda)$3

in a real codimension-$Co_0=O(\Lambda)$4 subspace. A hyperplane-arrangement argument yields a point

$Co_0=O(\Lambda)$5

which lies on no other reflection hyperplane $Co_0=O(\Lambda)$6 with $Co_0=O(\Lambda)$7 and $Co_0=O(\Lambda)$8. Choosing a small neighborhood $Co_0=O(\Lambda)$9 of $24$0 in $24$1 containing no other walls except $24$2, one obtains a decomposition

$24$3

Since both $24$4 and $24$5 meet $24$6, one may choose the unique Conway chamber $24$7 containing one side. Then $24$8 is a wall of $24$9, so $(G,S)$0 or $(G,S)$1 is a Leech-root for $(G,S)$2, and $(G,S)$3.

The Weyl-vector formalism then gives a primitive null vector $(G,S)$4 with

$(G,S)$5

and

$(G,S)$6

the hyperbolic plane. Since $(G,S)$7, it follows that $(G,S)$8 embeds primitively into $(G,S)$9. Because $S$0 fixes $S$1 and acts trivially on $S$2, it extends to $S$3 fixing $S$4 pointwise; then $S$5 preserves $S$6 and $S$7, hence fixes $S$8, and therefore acts trivially on the new orthogonal complement in $S$9.

The need for this refined chamber selection is not formal. Marquand–Muller exhibit a specific Leech pair $(G,S)$00 with $(G,S)$01 and $(G,S)$02, so that $(G,S)$03, for which the sketch in [GHV12] produces an embedding of $(G,S)$04 into $(G,S)$05 that fails to be primitive (Zheng, 14 Jul 2025). This demonstrates that the original proof is incomplete and that primitiveness is the subtle point.

4. Geometric role in symplectic automorphism problems

Leech pairs $(G,S)$06 and the embedding lemma play a central role in the classification of finite symplectic automorphism groups in hyperkähler and related geometries (Zheng, 14 Jul 2025). The applications listed in the source include the following:

• Symplectic automorphisms of $(G,S)$07-type manifolds: Mongardi–Huybrechts et al.
• Symplectic automorphisms of smooth cubic fourfolds: Laza–Zheng.
• Symplectic birational transformations of O’Grady $(G,S)$08 manifolds: Marquand–Muller.
• Symplectic automorphisms of $(G,S)$09 surfaces in positive characteristic: Ogus–Schütt, Wang.

In these settings one associates to a group $(G,S)$10 acting symplectically on the weight-two Hodge structure an invariant sublattice $(G,S)$11 with $(G,S)$12 and trivial $(G,S)$13-action, hence forming a Leech pair. The embedding lemma then yields an embedding

$(G,S)$14

so that $(G,S)$15 embeds in the Conway group

$(G,S)$16

Coupled with the Höhn–Mason classification of saturated subpairs of $(G,S)$17, this often yields a finite list of possibilities for $(G,S)$18 and $(G,S)$19, leading to a complete classification (Jiang et al., 2016).

A plausible implication is that the lattice-theoretic definition is designed not merely to encode a finite group action, but to isolate precisely the data that can be exported into the Leech-lattice environment where Conway-group methods become available.

5. The VOA-related definition $(G,S)$20

In the Lam–Miyamoto framework, the term Leech pair has a different meaning. Let $(G,S)$21 be the Leech lattice, $(G,S)$22 its lattice VOA, and let

$(G,S)$23

be an automorphism of $(G,S)$24, where $(G,S)$25, $(G,S)$26, and $(G,S)$27 is a fixed standard lift of $(G,S)$28. Following Möller–Scheithauer, such a $(G,S)$29 is called a generalized deep hole of $(G,S)$30 if the orbifold

$(G,S)$31

is a holomorphic VOA with non-zero weight-one space $(G,S)$32; equivalently, $(G,S)$33 is of type $(G,S)$34 in the sense that the $(G,S)$35-lowest weight of $(G,S)$36 is $(G,S)$37 (Lam et al., 2022).

The first lattice-theoretic facts are the existence of a canonical $(G,S)$38-duality isometry

$(G,S)$39

where $(G,S)$40 is the $(G,S)$41-coinvariant sublattice and $(G,S)$42 or $(G,S)$43, and the fact that pulling $(G,S)$44 through $(G,S)$45 produces an honest deep hole $(G,S)$46 which is $(G,S)$47-invariant and satisfies $(G,S)$48 (Lam et al., 2022).

A Leech pair is then a pair $(G,S)$49 satisfying three conditions. First,

$(G,S)$50

$(G,S)$51, $(G,S)$52 is a deep hole of $(G,S)$53, and $(G,S)$54. Second, if

$(G,S)$55

then its Coxeter number $(G,S)$56 is divisible by $(G,S)$57. Third, the root sublattice

$(G,S)$58

is, via glue code, the Niemeier root lattice $(G,S)$59 for some codeword $(G,S)$60 (Lam et al., 2022).

Two such Leech pairs $(G,S)$61 and $(G,S)$62 are equivalent, written

$(G,S)$63

if there exists $(G,S)$64 and $(G,S)$65 with

$(G,S)$66

so that $(G,S)$67 and $(G,S)$68 are equivalent deep holes, and if $(G,S)$69 and $(G,S)$70 induce the same isometry on the corresponding Niemeier lattice.

This definition packages orbifold data into a lattice-theoretic object built from $(G,S)$71-coinvariants, a deep hole, and a Niemeier neighbor. The explicit use of glue codes and Niemeier root lattices makes the construction combinatorial as well as geometric.

6. Coinvariant lattices, Niemeier neighbors, and the VOA classification theorem

For each $(G,S)$72 with frame shape $(G,S)$73, the coinvariant lattice $(G,S)$74 is constructed by the generalized Construction B. One takes

$(G,S)$75

lets $(G,S)$76 be the cyclic subgroup generated by one codeword $(G,S)$77, defines

$(G,S)$78

and

$(G,S)$79

and then checks that

$(G,S)$80

The same codeword $(G,S)$81 appears as the glue defining the Niemeier root lattice

$(G,S)$82

in condition (C3). Moreover, $(G,S)$83-duality identifies $(G,S)$84 with $(G,S)$85 up to scale $(G,S)$86, carries the dual lattice of the weight-one lattice $(G,S)$87 to the neighbor $(G,S)$88, and yields

$(G,S)$89

(Lam et al., 2022).

The main classification statement is Theorem 7.1: there is a one-to-one correspondence

$(G,S)$90

In the forward direction, given such a $(G,S)$91, one chooses a $(G,S)$92-element $(G,S)$93, forms the inner orbifold $(G,S)$94 giving $(G,S)$95, and obtains a reverse automorphism

$(G,S)$96

Then $(G,S)$97 satisfies (C1)–(C3), and changing $(G,S)$98 or its conjugate changes the pair only up to the equivalence relation above. Conversely, any Leech pair defines $(G,S)$99, and its fixed-point orbifold is a holomorphic $G$00 whose weight-one root system is exactly the scaled root system extracted from

$G$01

via $G$02-duality (Lam et al., 2022).

For each component $G$03 of $G$04, one finds its roots in $G$05 and hence, under $G$06, as a root system $G$07 of vectors of squared norm $G$08 in the affine layers $G$09. The quotient of $G$10 by $G$11 yields a finite Dynkin diagram whose type and level match exactly one of the $G$12 semisimple Lie algebras in Schellekens’ list. Table 2 describes, for each $G$13-class $G$14, how the cyclic diagram automorphism on the affine Niemeier root diagram collapses to the finite root diagram of each simple factor. Section 8 identifies the glue vectors

$G$15

as exactly the codewords that Höhn observed to parametrize these $G$16 cases. In this way one obtains a purely combinatorial proof of Schellekens’ classification, recovering not only the types of the $G$17 Lie algebras but also their levels via the $G$18-duality formula, and giving a uniform lattice-theoretic construction of all $G$19 non-trivial cases (Lam et al., 2022).

Taken together, the two notions of Leech pair show how the Leech lattice functions as a terminal object for disparate classification problems. In one direction, a finite-group action on a rootless lattice is forced into $G$20 through a primitive embedding theorem. In the other, an orbifold datum for $G$21 is encoded by a deep-hole/Niemeier pair and classified through $G$22-equivariant lattice combinatorics. A potential source of confusion is that the two definitions are not interchangeable; however, both are organized around the same structural principle: data with trivial fixed directions and controlled discriminant behavior can be transferred to the Leech lattice, where Conway-group and Niemeier-lattice methods become decisive.