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In this paper, we study the family of vertex operator algebras $SF(d)^+$,known as symplectic fermions. This family is of a particular interest becausethese VOAs are irrational and $C_2$-cofinite.  We determine the Zhu's algebra $A(SF(d)^+)$ and show that the equality ofdimensions of $A(SF(d)^+)$ and the $C_2$--algebra $\mathcal P(SF(d)^+)$ holdsfor $d \geq 2$ (the case of $d=1$ was treated by T. Abe). We use these resultsto prove a conjecture by Y. Arike and K. Nagatomo on the dimension of the spaceof one-point functions on $SF(d)^+$.

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