Classification of vertex operator algebras of class S⁴ with minimal conformal weight one

This is the authors' abstract. We don't add key points for this paper.

In this article, we describe the trace formulae of composition of several (up to four) adjoint actions of elements of the Lie algebra of a vertex operator algebra by using the Casimir elements. As an application, we give constraints on the central charge and the dimension of the Lie algebra for vertex operator algebras of class S⁴. In addition, we classify vertex operator algebras of class S⁴ with minimal conformal weight one under some assumptions.