INERTIA GROUPS AND UNIQUENESS OF HOLOMORPHIC VERTEX OPERATOR ALGEBRAS
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Springer Science+Business Media New York (2020)
INERTIA GROUPS AND UNIQUENESS OF HOLOMORPHIC VERTEX OPERATOR ALGEBRAS CHING HUNG LAM∗
HIROKI SHIMAKURA∗∗,∗∗∗
Institute of Mathematics Academia Sinica and National Center for Theoretical Sciences of Taiwan Taipei 10617, Taiwan
Graduate School of Information Sciences Tohoku University Sendai 980-8579, Japan [email protected]
[email protected]
Abstract. We continue our program on classification of holomorphic vertex operator algebras of central charge 24. In this article, we show that there exists a unique strongly regular holomorphic VOA of central charge 24, up to isomorphism, if its weight one Lie algebra has the type C4,10 , D7,3 A3,1 G2,1 , A5,6 C2,3 A1,2 , A3,1 C7,2 , D5,4 C3,2 A21,1 , or E6,4 C2,1 A2,1 . As a consequence, we have verified that the isomorphism class of a strongly regular holomorphic vertex operator algebra of central charge 24 is determined by its weight one Lie algebra structure if the weight one subspace is nonzero.
Contents 1. Introduction 2. Preliminary 2.1 Vertex operator algebras and weight one Lie algebras 2.2 ∆-operator, simple affine VOAs and twisted modules 3. Representation theory of simple current extensions 3.1 Simple current extensions and induced modules 3.2 Extension of LE6 (3, 0) 3.3 Extension of LA4 (5, 0) 3.4 Extension of LC5 (3, 0) ⊗ LA1 (1, 0) DOI: 10.1007/S00031-020-09570-8 C. H. Lam was partially supported by a research grant AS-IA-107-M02 of Academia Sinica and MoST grant 104-2115-M-001-004-MY3 of Taiwan. ∗∗ H. Shimakura was partially supported by JSPS KAKENHI Grant Numbers 26800001 and 17K05154. ∗∗∗ C. H. Lam and H. Shimakura were partially supported by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Development of Concentrated Mathematical Center Linking to Wisdom of the Next Generation”. Received August 20, 2018. Accepted July 14, 2019. Corresponding Author: Ching Hung Lam, e-mail: [email protected] ∗
CHING HUNG LAM, HIROKI SHIMAKURA
3.5 Extension of LA7 (4, 0) 4. Z2 -orbifold construction and the uniqueness of a holomorphic VOA 5. Quantum dimensions and mirror extensions 5.1 Quantum dimensions 5.2 Mirror extensions 5.3 Embedding of LA7 (4, 0) ⊗ LA3 (8, 0) in LA31 (1, 0) 6. Inertia groups 6.1 Definition of the inertia group 6.2 Weight one Lie algebra of type 6.3 Weight one Lie algebra of type 6.4 Weight one Lie algebra of type 6.5 Weight one Lie algebra of type
E6,3 G32,1 A24,5 C5,3 G2,2 A1,1 A7,4 A31,1
7. Orbifold constructions associated with inner automorphisms 7.1 General method 7.2 Case D7,3 A3,1 G2,1 7.3 Case D5,4 C3,2 A21,1 8. Main theorem A. Tables for highest weights 1. Introduction The classification of (strongly regular) holomorphic vertex operator algebras (VOAs) of central charge 24 is one of the important problems in VOA theory. In 1993, Schellekens [Sc93] obtained a list of 71 possible Lie algebra structures for the weight one subspace of a holomorphic VOA of central charge 24. It is also believed that the i
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