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

TL;DR

This paper constructs vertex operator algebras with arbitrary complex central charge whose Griess algebra is isomorphic to the symmetric matrices Jordan algebra, revealing how central charge deformations affect algebraic structures.

Contribution

It introduces a method to deform the central charge of vertex operator algebras while preserving a specific Griess algebra structure, specifically the Jordan algebra of symmetric matrices.

Findings

Constructed VOAs with arbitrary complex central charge

Griess algebra isomorphic to symmetric matrices Jordan algebra

Demonstrated deformation of central charges impacts algebraic properties

Abstract

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 $Sym_d(\C)$ 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 $Sym_d(\C)$ for any complex number $c$.