On Zhu’s algebra and $C_2$–algebra for symplectic fermion vertex algebra $SF(d)^+$

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 $SF(d)^+$ , known as symplectic fermions. This family is of a particular interest because these VOAs are irrational and $C_2$ -cofinite. We determine Zhu’s algebra $A(SF(d)^+)$ and show that the equality of dimensions of $A(SF(d)^+)$ and the $C_2$ -algebra $\mathcal{P}(SF(d)^+)$ holds for $d \geq 2$ (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 $SF(d)^+$ .

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