71 holomorphic vertex operator algebras of central charge 24

Lam, Ching Hung; Shimakura, Hiroki

doi:10.21915/bimas.2019105

Cited by 8 publications

(4 citation statements)

References 42 publications

(117 reference statements)

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“…Giving a rigorous proof of this has turned out to be a major project in its own right though by now it is done -except for the uniqueness of the Moonshine module V ♮ -thanks to the work of many. A good survey of this undertaking can be found in the paper of Lam and Shimakura [59].…”

Section: Holomorphic Voasmentioning

confidence: 99%

“…Schellekens showed why one should expect exactly 71 values of m that correspond to holomorphic VOAs, and, except possibly for the monstrous case m = −744 which remains open, it turns out that there is a unique such VOA for each of these values of m. The rigorous proof of this difficult result is due to many mathematicians, too many to cite here. For a good survey see the paper [59] of Lam and Shimakura. Throughout all of these works the subspace of the VOA of conformal weight 1 has the structure of a reductive Lie algebra, and the classification of such algebras plays, as it does in much of VOA theory, a major rôle. 3 It is a remarkable classification, particularly as any integer m ≥ −744 yields a plausible choice j + m for a character of a holomorphic VOA, yet most such values of m do not correspond to any holomorphic VOAs of central charge 24!…”

Section: Introductionmentioning

confidence: 99%

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Character Vectors of Strongly Regular Vertex Operator Algebras

Franc¹,

Mason²

2022

SIGMA

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We summarize interactions between vertex operator algebras and number theory through the lens of Zhu theory. The paper begins by recalling basic facts on vertex operator algebras (VOAs) and modular forms, and then explains Zhu's theorem on characters of VOAs in a slightly new form. We then axiomatize the desirable properties of modular forms that have played a role in Zhu's theorem and related classification results of VOAs. After this we summarize known classification results in rank two, emphasizing the geometric theory of vector-valued modular forms as a means for simplifying the discussion. We conclude by summarizing some known examples, and by providing some new examples, in higher ranks. In particular, the paper contains a number of potential character vectors that could plausibly correspond to a VOA, but such that the existence of a corresponding hypothetical VOA is presently unknown.

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Section: Holomorphic Voasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Character Vectors of Strongly Regular Vertex Operator Algebras

Franc¹,

Mason²

2022

SIGMA

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“…Item (a) was proved in [Sc93,EMS20] (see [ELMS21] for another proof). Items (b) and (c) were proved by using case by case analysis (see [LS19] and [LS20b,Introduction]); several uniform approaches for (b) and (c) are also discussed in [Hö,MS22+,MS,HM,CLM22].…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups and uniqueness of holomorphic vertex operator algebras of central charge $24$

Betsumiya¹,

Lam²,

Shimakura³

2022

Preprint

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We describe the automorphism groups of all holomorphic vertex operator algebras of central charge 24 with non-trivial weight one Lie algebras by using their constructions as simple current extensions. We also confirm a conjecture of G. Höhn on the numbers of holomorphic vertex operator algebras of central charge 24 obtained as inequivalent simple current extensions of certain vertex operator algebras, which gives another proof of the uniqueness of holomorphic vertex operator algebras of central charge 24 with non-trivial weight one Lie algebras.

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“…Since chiral conformal field theories are not deformable, their moduli space is discrete. For example, it has been predicted by physics that the moduli space of modular invariant chiral conformal field theories with central charge (24, 0) consists of 71 VOAs [S], which has been studied by mathematicians (see for example [DM,LS3]).…”

Section: Introductionmentioning

confidence: 99%

Code conformal field theory and framed algebra

Moriwaki¹

2021

Preprint

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Let l, r ∈ Z ≥0 . We mathematically study conformal field theories with central charge ( l 2 , r 2 ) which have the symmetry of the (l + r) Virasoro algebras of central charge 1 2 , Vir ⊕lin its holomorphic and anti-holomorphic parts. Such conformal field theories are called (l, r)-code conformal field theories in this paper. We introduce a notion of an (l, r)-framed algebra which is a non-associative finite dimensional algebra and show that the category of (l, r)-code conformal field theories and the category of (l, r)-framed algebras are equivalent. Therefore, the construction of code CFTs are reduced to the construction of framed algebras.For each code G ⊂ Z r 2 satisfying 1 r = (1, 1, . . . , 1) ∈ G, we constructed an (r, r)-framed algebra S G . The corresponding code CFT F G defines a modular invariant conformal field theory. For example, when G = 1 r , the corresponding conformal field theory is the SO(r)-WZW model at level 1 for r ≥ 4 and the critical Ising model (resp. the SO(3)-WZW model at level 2) for r = 1 (resp. r = 3). We can also obtain a family of new conformal field theories by considering nontrivial codes G. All operator product expansions and the four-point correlation functions of these conformal field theories can be computed combinatorially using the codes. Furthermore, by considering the current-current deformation of code conformal field theories, we mathematically construct a continuous family of conformal field theories.

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71 holomorphic vertex operator algebras of central charge 24

Cited by 8 publications

References 42 publications

Character Vectors of Strongly Regular Vertex Operator Algebras

Character Vectors of Strongly Regular Vertex Operator Algebras

Automorphism groups and uniqueness of holomorphic vertex operator algebras of central charge $24$

Code conformal field theory and framed algebra

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