Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra

Möller, Sven; Scheithauer, Nils R.

doi:10.48550/arxiv.1910.04947

Cited by 9 publications

(84 citation statements)

References 24 publications

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“…Then, using geometric arguments similar to those by Conway, Parker and Sloane in [CPS82] we reduce this number to 70. In [MS19] we explicitly list 70 generalised deep holes with different diagrams so that there are exactly these diagrams.…”

Section: Theorem (Classification Of Generalised Deep Holes)mentioning

confidence: 99%

“…Automorphisms of the Leech Lattice Vertex Operator Algebra. We describe lattice vertex operator algebras [Bor86,FLM88], the automorphism group of the Leech lattice vertex operator algebra V Λ and in particular its conjugacy classes, which were determined in [MS19].…”

Section: Vertex Operator Algebras and Their Automorphismsmentioning

confidence: 99%

“…In this paper we give a new, geometric proof of the theorem based on the results in [MS19], which generalises the classification of the Niemeier lattices by enumeration of the corresponding deep holes of the Leech lattice Λ [CPS82,Bor85].…”

mentioning

confidence: 99%

“…Suppose that V has central charge 24 and that n > 1. Pairing the character of V g with a certain vector-valued Eisenstein series of weight 2 we obtain [MS19]: Theorem (Dimension Formula). The dimension of the weight-1 subspace of V orb(g) is given by dim(V orb(g) 1…”

mentioning

confidence: 99%

“…Recall that algebraic conjugacy means conjugacy of cyclic subgroups. An averaged version of Kac's very strange formula implies [MS19]:…”

mentioning

confidence: 99%

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A Geometric Classification of the Holomorphic Vertex Operator Algebras of Central Charge 24

Möller¹,

Scheithauer²

2021

Preprint

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We associate with a generalised deep hole of the Leech lattice vertex operator algebra a generalised hole diagram. We show that this Dynkin diagram determines the generalised deep hole up to conjugacy and that there are exactly 70 such diagrams. In an earlier work we proved a bijection between the generalised deep holes and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-trivial weight-1 space. Hence, we obtain a new, geometric classification of these vertex operator algebras. Contents

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Section: Theorem (Classification Of Generalised Deep Holes)mentioning

confidence: 99%

Section: Vertex Operator Algebras and Their Automorphismsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Recall that algebraic conjugacy means conjugacy of cyclic subgroups. An averaged version of Kac's very strange formula implies [MS19]:…”

mentioning

confidence: 99%

See 3 more Smart Citations

A Geometric Classification of the Holomorphic Vertex Operator Algebras of Central Charge 24

Möller¹,

Scheithauer²

2021

Preprint

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A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra

Lam,

Miyamoto

2023

Forum of Mathematics, Sigma

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We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_\Lambda $ . We show that a generalized deep hole defines a ‘true’ automorphism invariant deep hole of the Leech lattice. We also show that there is a correspondence between the set of isomorphism classes of holomorphic VOA V of central charge $24$ having non-abelian $V_1$ and the set of equivalence classes of pairs $(\tau , \tilde {\beta })$ satisfying certain conditions, where $\tau \in Co.0$ and $\tilde {\beta }$ is a $\tau $ -invariant deep hole of squared length $2$ . It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge $24$ . In particular, we give an explanation for an observation of G. Höhn, which relates the weight one Lie algebras of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices.

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Systematic orbifold constructions of Schellekens' vertex operator algebras from Niemeier lattices

Höhn

Möller

2022

Journal of London Math Soc

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We present a systematic, rigorous construction of all 70 strongly rational, holomorphic vertex operator algebras V$V$ of central charge 24 with non‐zero weight‐one space V1$V_1$ as cyclic orbifold constructions associated with the 24 Niemeier lattice vertex operator algebras VN$V_N$ and certain 226 short automorphisms in prefixAutfalse(VNfalse)$\operatorname{Aut}(V_N)$. We show that up to algebraic conjugacy these automorphisms are exactly the generalised deep holes, as introduced in Möller and Scheithauer (Ann. of Math. (to appear)), of the Niemeier lattice vertex operator algebras with the additional property that their orders are equal to those of the corresponding outer automorphisms. Together with the constructions in Höhn (2017) and Möller and Scheithauer (Ann. of Math. (to appear)), this gives three different uniform constructions of these vertex operator algebras, which are related through 11 algebraic conjugacy classes in prefixCo0$\operatorname{Co}_0$. Finally, by considering the inverse orbifold constructions associated with the 226 short automorphisms, we give the first systematic proof of the result that each strongly rational, holomorphic vertex operator algebra V$V$ of central charge 24 with non‐zero weight‐one space V1$V_1$ is uniquely determined by the Lie algebra structure of V1$V_1$.

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Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra

Cited by 9 publications

References 24 publications

A Geometric Classification of the Holomorphic Vertex Operator Algebras of Central Charge 24

A Geometric Classification of the Holomorphic Vertex Operator Algebras of Central Charge 24

A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra

Systematic orbifold constructions of Schellekens' vertex operator algebras from Niemeier lattices

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