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On Zhu’s algebra and \(C_2\)-algebra for symplectic fermion vertex algebra \(SF (d)^+\). (English) Zbl 07242589
Summary: 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 \(SF (d)^+\).
MSC:
17B69 Vertex operators; vertex operator algebras and related structures
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