**Authors:** Dražen Adamović, Ante Čeperić

Journal of Algebra, Volume 563, 1 December 2020, Pages 376-403, https://doi.org/10.1016/j.jalgebra.2020.07.019

**Abstract:** In this paper, we study the family of vertex operator algebras , known as symplectic fermions. This family is of a particular interest because these VOAs are irrational and -cofinite. We determine Zhu’s algebra and show that the equality of dimensions of and the -algebra 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 .

**Keywords:** Vertex algebra, Zhu’s algebra, Symplectic fermions