Lattice Subalgebras of Strongly Regular Vertex Operator Algebras

Mason, Geoffrey

doi:10.1007/978-3-662-43831-2_2

Cited by 21 publications

(29 citation statements)

References 60 publications

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“…For a review of the theory of such vertex operator algebras, cf. [26]. A simple, strongly regular VOA satisfies the following additional properties (loc.…”

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confidence: 99%

Jacobi trace functions in the theory of vertex operator algebras

Krauel

Mason

2015

Communications in Number Theory and Physics

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We describe a type of n-point function associated to strongly regular vertex operator algebras V and their irreducible modules. Transformation laws with respect to the Jacobi group are developed for 1-point functions. For certain elements in V , the finite-dimensional space spanned by the 1-point functions for the irreducible modules is shown to be a vector-valued weak Jacobi form. A decomposition of 1-point functions for general elements is proved, and shows that such functions are typically quasi-Jacobi forms. Zhu-type recursion formulas are provided; they show how an n-point function can be written as a linear combination of (n − 1)-point functions with coefficients that are quasi-Jacobi forms.

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“…For a review of the theory of such vertex operator algebras, cf. [26]. A simple, strongly regular VOA satisfies the following additional properties (loc.…”

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confidence: 99%

Jacobi trace functions in the theory of vertex operator algebras

Krauel

Mason

2015

Communications in Number Theory and Physics

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“…If V is a rational C 2 -cofinite simple VOA of strong CFT type, then J 1 (V) = 0. This follows immediately from [Ma,Theorem 3]. Hence, if V is a rational C 2 -cofinite simple unitary VOA of strong CFT type with an anti-involution σ, then the vertex algebra automorphism group of V σ coincides with the VOA automorphism group.…”

Section: Vertex Operator Algebra With Positive-definite Invariant Formmentioning

confidence: 82%

On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

Moriwaki

2020

Journal of Algebra

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Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automorphism group. Furthermore, we prove that the full vertex algebra automorphism group of the moonshine module over the field of real numbers is the Monster.

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“…It is shown in [DM04b] and [Mas14], Section 3.3, that every V -module is a completely reducible V 1 -module. Then if v ∈ V 1 is a semisimple element, the zero mode v 0 acts semisimply on every V -module (and in particular on V itself).…”

Section: Concrete Approachmentioning

confidence: 99%

Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

Ekeren

Möller

Scheithauer

2018

International Mathematics Research Notices

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We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order n such that Γ 0 (n) is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6 and 8 to establish that each of the following fifteen Lie algebras is the weight-one space V 1 of exactly one holomorphic, C 2 -cofinite vertex operator algebra V of CFT-type and central charge 24: 3,4 , corresponding to the cases 44, 33, 36, 62, 26, 52, 22, 13, 53, 48, 9, 40, 56, 21 and 7 on Schellekens' list.The strategy we use to prove uniqueness is developed in [LS16c] and is based in an essential way on the inverse orbifold construction, first developed in [EMS15] (and called "reverse orbifold" in [LS16c]). In a nutshell, the idea is as follows: Let V be a strongly rational, holomorphic vertex operator algebra of central charge 24 with weight-one Lie algebra V 1 . Any inner automorphism σ = e (2πi) adv of V 1 , v ∈ V 1 , extends to an automorphism of V . Choose such an automorphism, and suppose that V orb(σ) is isomorphic to the lattice vertex operator algebra V L . By virtue of its construction as an orbifold, V orb(σ) carries an inverse orbifold automorphism ζ of the same order as σ with V = (V orb(σ) ) orb(ζ) . Under favourable circumstances the order and the fixed-point subalgebra (Outline. We assume that the reader is familiar with vertex operator algebras and their representation theory and with the specific examples provided by lattice vertex operator algebras and affine vertex operator algebras (see, e.g. [LL04]).The article is organised as follows: In Section 2 we describe Lie algebras occurring as weight-one spaces of vertex operator algebras, inner automorphisms and the classification of the weight-one structures of strongly rational, holomorphic vertex operator algebras of central charge 24, dubbed Schellekens' list.Section 3 reviews the cyclic orbifold construction of strongly rational, holomorphic vertex operator algebras and the inverse orbifold construction. In Section 4 a genus zero dimension formula for the weight-one space of these orbifolds for central charge 24 is derived.In Section 5 the general procedure to prove the uniqueness of certain vertex operator algebras on Schellekens' list is described. This is split up into three parts, which are described in detail in Sections 6, 7 and 8 for the fifteen cases considered in this text.Finally, in Section 9 we state the main result of this paper.Acknowledgements. The authors would like to thank Ching Hung Lam and Hiroki Shimakura for helpful discussions. The second and third author both were partially supported by the DFG project "Infinite-dimensional Lie algebras in string theory". A part of the work was done while the first and second author visited the ESI in Vienna for the conference "Geometry and Representation Theory" in 2017.They are grateful to both the ESI and to the organisers of the conference. We thank the two anonymous referees for their comments and suggestions....

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Lattice Subalgebras of Strongly Regular Vertex Operator Algebras

Cited by 21 publications

References 60 publications

Jacobi trace functions in the theory of vertex operator algebras

Jacobi trace functions in the theory of vertex operator algebras

On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

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