A holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra \(F_{4,6}A_{2,2}\). (English) Zbl 1469.17026
Summary: In this paper, a holomorphic vertex operator algebra \(U\) of central charge 24 with the weight one Lie algebra \(A_{8, 3} A_{2, 1}^2\) is proved to be unique. Moreover, a holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra \(F_{4, 6} A_{2, 2}\) is obtained by applying a \(\mathbb{Z}_2\)-orbifold construction to \(U\). The uniqueness of such a vertex operator algebra is also established. By a similar method, we also established the uniqueness of a holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra \(E_{7, 3} A_{5, 1}\). As a consequence, we verify that all 71 Lie algebras in Schellekens’ list can be realized as the weight one Lie algebras of some holomorphic vertex operator algebras of central charge 24. In addition, we establish the uniqueness of three holomorphic vertex operator algebras of central charge 24 whose weight one Lie algebras have the type \(A_{8, 3} A_{2, 1}^2, F_{4, 6} A_{2, 2}\), and \(E_{7, 3} A_{5, 1}\).
MSC:
| 17B69 | Vertex operators; vertex operator algebras and related structures |
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