On the Genus of the Moonshine Module

Höhn, Gerald

doi:10.48550/arxiv.1708.05990

Cited by 12 publications

(70 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We remark that Höhn's approach to the classification problem in [Höh17] (and [Lam20]) based on coset constructions can in principle also be used to give a uniform proof of the above classification result.…”

Section: Theorem (Classification Of Generalised Deep Holes)mentioning

confidence: 95%

“…This is the decomposition of the genus of the Moonshine module described by Höhn in [Höh17]. The connection is explored in [HM20].…”

Section: Theorem (Classification Of Generalised Deep Holes)mentioning

confidence: 99%

“…Under the natural projection Aut(V Λ ) → O(Λ) the generalised deep holes g of V Λ with rk((V g Λ ) 1 ) > 0 map to the 11 algebraic conjugacy classes in O(Λ) ∼ = Co 0 with cycle shapes 1 24 , 1 8 2 8 , 1 6 3 6 , 2 12 , 1 4 2 2 4 4 , 1 4 5 4 , 1 2 2 2 3 2 6 2 , 1 3 7 3 , 1 2 2 1 4 1 8 2 , 2 3 6 3 and 2 2 10 2 . This is the decomposition of the genus of the Moonshine module described by Höhn in [Höh17].…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Möller¹,

Scheithauer²

2021

Preprint

View full text Add to dashboard Cite

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

show abstract

Section: Theorem (Classification Of Generalised Deep Holes)mentioning

confidence: 95%

“…This is the decomposition of the genus of the Moonshine module described by Höhn in [Höh17]. The connection is explored in [HM20].…”

Section: Theorem (Classification Of Generalised Deep Holes)mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

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

Möller¹,

Scheithauer²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The classification of holomorphic vertex operator algebras (VOAs) of central charge 24 has been completed except for the uniqueness of the moonshine VOA, and several uniform proofs are proposed (see [Hö,MS,HM,ELMS21,CLM] and the references therein). One of them, proposed by Höhn in [Hö], is to view a holomorphic VOA V of central charge 24 with V 1 = 0 as a simple current extension of the tensor product VOA V Lg ⊗ V ĝ Λg .…”

Section: Introductionmentioning

confidence: 99%

“…Here V Lg is the lattice VOA associated with a lattice L g related to the root lattice of g = V 1 , g is some isometry of the Leech lattice Λ and V ĝ Λg is the cyclic orbifold VOA associated with the coinvariant lattice Λ g and a lift ĝ ∈ Aut (V Λg ) of the restriction of g to Λ g . Höhn also described the possible isometries g; there are 10 different cases [Hö,Table 4].…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

Betsumiya¹,

Lam²,

Shimakura³

2021

Preprint

View full text Add to dashboard Cite

We determine the automorphism groups of the cyclic orbifold vertex operator algebras associated with coinvariant lattices of isometries of the Leech lattice in the conjugacy classes 4C, 6E, 6G, 8E and 10F . As a consequence, we have determined the automorphism groups of all the 10 vertex operator algebras in [Hö], which are useful to analyze holomorphic vertex operator algebras of central charge 24.

show abstract

Cyclic orbifolds of lattice vertex operator algebras having group-like fusions

Lam

2019

Lett Math Phys

View full text Add to dashboard Cite

Let L be an even (positive definite) lattice and g ∈ O(L). In this article, we prove that the orbifold vertex operator algebra Vĝ L has group-like fusion if and only if g acts trivially on the discriminant group D(L) = L * /L (or equivalently (1 − g)L * < L).We also determine their fusion rings and the corresponding quadratic space structures when g is fixed point free on L. By applying our method to some coinvariant sublattices of the Leech lattice Λ, we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge 24 using the the orbifold vertex operator algebra Vĝ Λg .where wt(·) denotes the conformal weight of the module. Moreover, R(V Lg ) is isomorphic to D(L g ) = L * g /L g as a quadratic space. By some recent results on coset constructions [CKM1, Main Theorem 2] (see also[KMi15] and [Lin17]), it is known that the VOA W also has group-like fusion and R(W ) forms a quadratic space isomorphic to (R(V Lg ), −q), where the quadratic form is defined by conformal weights modulo Z. Since V Lg ⊗ W is a full subVOA of V and V is holomorphic, the VOA V defines a maximal totally singular subspace of R(V Lg ) × R(W ); hence it induces an anti-isomorphism of quadratic spaces ϕ :2 Conversely, let ϕ : (R(V Lg ), q) → (R(W ), −q ′ ) be an isomorphism of quadratic spaces. Then the set {(M, ϕ(M)) | M ∈ R(V Lg )} is a maximal totally singular subspace of R(V Lg ) × R(W ) and hence U = M ∈R(V Lg ) M ⊗ ϕ(M) has a structure of a holomorphic VOA.

show abstract

On the Genus of the Moonshine Module

Cited by 12 publications

References 11 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

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

Cyclic orbifolds of lattice vertex operator algebras having group-like fusions

Contact Info

Product

Resources

About