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

TL;DR

This paper analyzes the structure of symplectic fermion vertex algebras, determining their Zhu's algebra and $C_2$-algebra, and confirms a conjecture about the dimension of one-point functions.

Contribution

It explicitly computes Zhu's algebra for $SF(d)^+$ and proves the dimension equality with the $C_2$-algebra for $d eq 1$, confirming a conjecture on one-point functions.

Findings

Zhu's algebra $A(SF(d)^+)$ is explicitly determined.

Dimension equality between $A(SF(d)^+)$ and $ ext{C}_2$-algebra holds for $d eq 1$.

Confirmed the conjecture on the dimension of one-point functions for $SF(d)^+$.

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 $d \geq 2$ (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 $SF(d)^+$.