On Zhu's algebra and C2–algebra for symplectic fermion vertex algebra SF(d)+

Abstract

In this paper, we study the family of vertex operator algebras SF(d)+, known as symplectic fermions. This family is of a particular interest because these VOAs are irrational and C2-cofinite. We determine Zhu's algebra A(SF(d)+) and show that the equality of dimensions of A(SF(d)+) and the C2–algebra P(SF(d)+) holds for d≥2 (the case of d=1 was treated by T. Abe in [1]). We use these results to prove a conjecture by Y. Arike and K. Nagatomo ([8]) on the dimension of the space of one-point functions on SF(d)+.