On Zhu's algebra and $C_2$--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 $C_2$-cofinite. We determine the 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 $d\ge 2$ (the case of $d=1$ was treated by T. Abe). We use these results to prove a conjecture by Y. Arike and K. Nagatomo on the dimension of the space of one-point functions on $SF(d{)}^{+}$.

25 pages