Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24

In 1993, Schellekens [‘Meromorphic c=24 conformal field theories’, Comm. Math. Phys. 153 (1993) 159–185.] obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras (VOAs) using the theory of framed VOAs and to determine the Lie algebra structures of their weight 1 subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes RM(1, 4)3 and RM(1, 4)⊕ 𝒟(d16+) of length 48. These VOAs correspond to the holomorphic simple current extensions of the lattice type VOAs ( V 2 E 8 + ) ⊗ 3 and V 2 E 8 + ⊗ V 2 D 16 + + . We determine such extensions using a quadratic space structure on the set of all irreducible modules R(W) of W when W = ( V 2 E 8 + ) ⊗ 3 or V 2 E 8 + ⊗ V 2 D 16 + + As our main results, we construct seven new holomorphic VOAs of central charge 24 in Schellekens' list and obtain a complete list of all Lie algebra structures associated to the weight 1 subspaces of holomorphic framed VOAs of central charge 24.