---
title: 'Leech Pair: Lattice and VOA Perspectives'
url: https://www.emergentmind.com/topics/leech-pair
type: topic
---

# Leech Pair: Lattice and VOA Perspectives

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" [2507.10414], 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 [2205.04681], 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 $r$. Its dual lattice is
$$
S^*:=\operatorname{Hom}(S,\mathbb{Z}),
$$
and its discriminant group is
$$
A_S:=S^*/S,
$$
an abelian group equipped with the natural $\mathbb{Q}/\mathbb{Z}$-valued quadratic form induced by $(\cdot,\cdot)$ on $S$. The quantity $\ell(A_S)$ is the minimal number of generators of $A_S$, referred to as its length [2507.10414].

A pair $(G,S)$ is called a Leech pair when the following conditions hold. First, $S$ is even, positive definite, and rootless, meaning that there is no $v\in S$ with $(v,v)=2$. Second, $G\le O(S)$ acts faithfully on $S$ and has no nonzero invariants:
$$
S^G:=\{x\in S\mid g\cdot x=x\ \text{for all}\ g\in G\}=\{0\}.
$$
Third, every $g\in G$ induces the identity on $A_S$, equivalently
$$
G\le \widetilde O(S):=\{\gamma\in O(S)\mid \gamma|_{A_S}=\mathrm{id}\}.
$$
The same condition can be stated by saying that $G$ fixes $S$ pointwise modulo $S$ and leaves no nonzero vector of $S$ invariant [2507.10414].

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 $A_S$ is the hypothesis that later permits extension of the group action across a primitive embedding.

## 2. The embedding lemma and the numerical bound

Let $\Lambda$ denote the Leech lattice, the unique even, positive-definite, unimodular, rootless lattice of rank $24$. The central structural statement is the corrected Gaberdiel–Hohenegger–Volpato primitive embedding lemma: if $(G,S)$ is a Leech pair and
$$
\operatorname{rank}(S)+\ell(A_S)\le 24,
$$
then there exists a primitive, $G$-equivariant embedding
$$
S\hookrightarrow \Lambda
$$
such that $G$ acts trivially on the orthogonal complement $S^\perp$ in $\Lambda$ [2507.10414].

In the formulation given in [2507.10414], the conclusion is stronger than mere existence of an embedding. The embedding is required to be primitive, the $G$-action must extend from $S$ to $\Lambda$, and the extension is constrained so that
$$
G\big|_{S^\perp}=\mathrm{id}.
$$
This makes the lemma a transfer principle from an abstract rootless lattice with controlled discriminant action into the universal rank-$24$ rootless unimodular setting.

The role of the bound $\operatorname{rank}(S)+\ell(A_S)\le 24$ is explicit. It is used to invoke Nikulin’s theorem on primitive embeddings into the even unimodular lattice of signature $(25,1)$. Without that numerical constraint, one cannot guarantee the existence of a primitive embedding of $S\oplus A_1$.

## 3. Corrected proof and the counterexample to the original argument

The corrected proof has two principal stages [2507.10414]. The first is an embedding into the even unimodular lattice
$$
II_{25,1}.
$$
By Nikulin’s theorem on primitive embeddings, the bound
$$
\operatorname{rank}(S)+\ell(A_S)\le 24
$$
guarantees a primitive orthogonal embedding
$$
S\oplus A_1\hookrightarrow II_{25,1},
$$
where $A_1=\langle \alpha\rangle$ is the rank-one lattice of norm $+2$.

The second stage passes from $II_{25,1}$ to the Leech lattice through a Conway chamber argument. Writing $L:=II_{25,1}$, one chooses the image $\alpha\in L$ of the $A_1$-generator and a component
$$
D\subset L\otimes \mathbb{R}
$$
of $\{x\mid (x,x)<0\}$, the Lobachevsky cone. One then studies
$$
S^\perp_D:=\{x\in D\mid (x,s)=0\ \text{for all}\ s\in S\},
$$
which has $\dim S^\perp_D\ge 2$ and meets the reflection hyperplane
$$
H_\alpha:=\{x\mid (x,\alpha)=0\}
$$
in a real codimension-$1$ subspace. A hyperplane-arrangement argument yields a point
$$
x\in S^\perp_D\cap H_\alpha
$$
which lies on no other reflection hyperplane $H_\beta$ with $\beta^2=2$ and $\beta\ne \pm \alpha$. Choosing a small neighborhood $U$ of $x$ in $D$ containing no other walls except $H_\alpha$, one obtains a decomposition
$$
U\setminus H_\alpha = U_1\sqcup U_2.
$$
Since both $U_1$ and $U_2$ meet $S^\perp_D$, one may choose the unique Conway chamber $C_w$ containing one side. Then $H_\alpha$ is a wall of $C_w$, so $\alpha$ or $-\alpha$ is a Leech-root for $w$, and $(\alpha,w)=-1$.

The Weyl-vector formalism then gives a primitive null vector $w\in L$ with
$$
w^\perp/\mathbb{Z}w \cong \Lambda,
$$
and
$$
\langle \alpha,w\rangle \cong U,
$$
the hyperbolic plane. Since $\alpha,w\perp S$, it follows that $S$ embeds primitively into $w^\perp/\mathbb{Z}w\cong \Lambda$. Because $G$ fixes $S$ and acts trivially on $A_S$, it extends to $L$ fixing $S^\perp$ pointwise; then $G$ preserves $S^\perp_D$ and $C_w$, hence fixes $w$, and therefore acts trivially on the new orthogonal complement in $\Lambda$.

The need for this refined chamber selection is not formal. Marquand–Muller exhibit a specific Leech pair $(G,S)$ with $\operatorname{rank}S=22$ and $\ell(A_S)=2$, so that $\operatorname{rank}+\ell=24$, for which the sketch in [GHV12] produces an embedding of $S$ into $\Lambda$ that fails to be primitive [2507.10414]. This demonstrates that the original proof is incomplete and that primitiveness is the subtle point.

## 4. Geometric role in symplectic automorphism problems

Leech pairs $(G,S)$ and the embedding lemma play a central role in the classification of finite symplectic automorphism groups in hyperkähler and related geometries [2507.10414]. The applications listed in the source include the following:

- **Symplectic automorphisms of $K3^{[n]}$-type manifolds**: Mongardi–Huybrechts et al.
- **Symplectic automorphisms of smooth cubic fourfolds**: Laza–Zheng.
- **Symplectic birational transformations of O’Grady $OG10$ manifolds**: Marquand–Muller.
- **Symplectic automorphisms of $K3$ surfaces in positive characteristic**: Ogus–Schütt, Wang.

In these settings one associates to a group $G$ acting symplectically on the weight-two Hodge structure an invariant sublattice $S$ with $S^G=0$ and trivial $A_S$-action, hence forming a Leech pair. The embedding lemma then yields an embedding
$$
S\hookrightarrow \Lambda,
$$
so that $G$ embeds in the Conway group
$$
Co_0=O(\Lambda).
$$
Coupled with the Höhn–Mason classification of saturated subpairs of $(Co_0,\Lambda)$, this often yields a finite list of possibilities for $G$ and $S$, leading to a complete classification [1604.04017].

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 $(\tau,\tilde\beta)$

In the Lam–Miyamoto framework, the term **Leech pair** has a different meaning. Let $\Lambda$ be the Leech lattice, $V_\Lambda$ its lattice VOA, and let
$$
g=\hat\tau\,\exp(2\pi i\,\beta(0))
$$
be an automorphism of $V_\Lambda$, where $\tau\in Co_0$, $\beta\in \mathbb{Q}\otimes \Lambda$, and $\hat\tau$ is a fixed standard lift of $\tau$. Following Möller–Scheithauer, such a $g$ is called a generalized deep hole of $V_\Lambda$ if the orbifold
$$
V=(V_\Lambda)^g\oplus \bigoplus_{j=1}^{n-1} V_\Lambda[g^j]
$$
is a holomorphic VOA with non-zero weight-one space $V_1$; equivalently, $g$ is of type $0$ in the sense that the $L_0$-lowest weight of $V_\Lambda[g]$ is $<1$ [2205.04681].

The first lattice-theoretic facts are the existence of a canonical $\ell$-duality isometry
$$
\varphi_T:(\Lambda_T)^*\to \Lambda_T,
$$
where $\Lambda_T=(1-\tau)\Lambda$ is the $\tau$-coinvariant sublattice and $\ell=[\tau]$ or $2[\tau]$, and the fact that pulling $\beta$ through $\varphi_T$ produces an honest deep hole $B\in \Lambda$ which is $\tau$-invariant and satisfies $(B,B)=2$ [2205.04681].

A Leech pair is then a pair $(\tau,\tilde\beta)$ satisfying three conditions. First,
$$
\tau\in P_0=\{1A,2A,2C,3B,4C,5B,6E,6G,7B,8E,10F\}\subset Co_0,
$$
$\tilde\beta\in (\Lambda_T)^*$, $B=\varphi_T(\tilde\beta)$ is a deep hole of $\Lambda$, and $(B,B)=2$. Second, if
$$
N=\Lambda_T+\mathbb{Z}B,
$$
then its Coxeter number $h$ is divisible by $|\tau|$. Third, the root sublattice
$$
R(N)=\langle \{v\in N\mid (v,v)=2\}\rangle
$$
is, via glue code, the Niemeier root lattice $L_A(c)$ for some codeword $c$ [2205.04681].

Two such Leech pairs $(\tau,\tilde\beta)$ and $(\tau',\tilde\beta')$ are equivalent, written
$$
(\tau,\tilde\beta)\sim (\tau',\tilde\beta'),
$$
if there exists $o\in Co_0$ and $\lambda\in \Lambda$ with
$$
\tilde\beta'=o(\tilde\beta-\lambda),
$$
so that $B$ and $B'$ are equivalent deep holes, and if $\tau'$ and $o\tau o^{-1}$ induce the same isometry on the corresponding Niemeier lattice.

This definition packages orbifold data into a lattice-theoretic object built from $\tau$-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 $\tau\in P_0$ with frame shape $\prod_m m^{k_m}$, the coinvariant lattice $\Lambda_T$ is constructed by the generalized Construction B. One takes
$$
R=\bigoplus_k A_{k-1},
$$
lets $C\subset \prod \mathbb{Z}/k_i\mathbb{Z}$ be the cyclic subgroup generated by one codeword $c$, defines
$$
L_A(C)=\{x\in R^*\mid x \bmod R\in C\},
$$
and
$$
L_B(C)=\{x\in L_A(C)\mid (x,\sum \text{fund roots})\in \mathbb{Z}\},
$$
and then checks that
$$
L_B(C)\cong \Lambda_T.
$$
The same codeword $c$ appears as the glue defining the Niemeier root lattice
$$
R(N)=L_A(c)
$$
in condition (C3). Moreover, $\ell$-duality identifies $(\Lambda_T)^*$ with $\Lambda_T$ up to scale $\ell$, carries the dual lattice of the weight-one lattice $L^*$ to the neighbor $N=\Lambda_T+\mathbb{Z}B$, and yields
$$
N_T\cong \varphi_T(L^*)
\qquad\text{and}\qquad
N=\Lambda_T+\mathbb{Z}B\ \text{a Niemeier lattice}
$$
[2205.04681].

The main classification statement is Theorem 7.1: there is a one-to-one correspondence
$$
\{\text{isomorphism classes of holomorphic VOAs }V,\ c=24,\ V_1\neq \text{abelian}\}
\leftrightarrow
\{\text{Leech pairs }(\tau,\tilde\beta)\text{ satisfying (C1)–(C3)}\}/\sim.
$$
In the forward direction, given such a $V$, one chooses a $W$-element $a\in V_1$, forms the inner orbifold $\exp(2\pi i\,a(0))$ giving $V_\Lambda$, and obtains a reverse automorphism
$$
g=\hat\tau\,\exp(2\pi i\,\beta(0)).
$$
Then $(\tau,\tilde\beta)$ satisfies (C1)–(C3), and changing $a$ or its conjugate changes the pair only up to the equivalence relation above. Conversely, any Leech pair defines $g$, and its fixed-point orbifold is a holomorphic $V$ whose weight-one root system is exactly the scaled root system extracted from
$$
N=\Lambda_T+\mathbb{Z}B
$$
via $\ell$-duality [2205.04681].

For each component $G_i$ of $V_1$, one finds its roots in $L^*$ and hence, under $\varphi_T$, as a root system $R_k$ of vectors of squared norm $2$ in the affine layers $N-kB$. The quotient of $R_k$ by $\tau$ yields a finite Dynkin diagram whose type and level match exactly one of the $69$ semisimple Lie algebras in Schellekens’ list. Table 2 describes, for each $P_0$-class $\tau$, 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
$$
c\in D(R(N))
$$
as exactly the codewords that Höhn observed to parametrize these $69$ cases. In this way one obtains a purely combinatorial proof of Schellekens’ classification, recovering not only the types of the $V_1$ Lie algebras but also their levels via the $\ell$-duality formula, and giving a uniform lattice-theoretic construction of all $70$ non-trivial cases [2205.04681].

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 $\Lambda$ through a primitive embedding theorem. In the other, an orbifold datum for $V_\Lambda$ is encoded by a deep-hole/Niemeier pair and classified through $Co_0$-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.

Source: https://www.emergentmind.com/topics/leech-pair