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

If a vertex operator algebra $V={\oplus }_{n=0}^{\infty }{V}_n$ satisfies $\dim V_0=1$, $V_1=0$, then $V_2$ 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 ${\mathrm{Sym}}_d(\mathbb{C})$ 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 ${\mathrm{Sym}}_d(\mathbb{C})$ for any complex number c and a positive integer d.