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

Authors: Dražen AdamovićAnte 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≥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