Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras

If a vertex operator algebra satisfies dimV0=1, V1=0, then V2 has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set Symd(C) of symmetric matrices of degree d becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge c and its Griess algebra is isomorphic to Symd(C) for any complex number c and a positive integer d.