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On parafermion vertex algebras of $𝔰 𝔩 (2)$ and $𝔰 𝔩 (3)$ at level $- \frac{3}{2}$

Dražen Adamović

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10 000 Zagreb, Croatia

E-mail Address: adamovic@math.hr

Corresponding author.

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Antun Milas

Department of Mathematics and Statistics, SUNY Albany, 1400 Washington Avenue, Albany, NY 12222, USA

E-mail Address: amilas@albany.edu

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, and

Qing Wang

School of Mathematical Sciences, Xiamen University, Fujian 361005, P. R. China

E-mail Address: qingwang@xmu.edu.cn

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https://doi.org/10.1142/S0219199720500868Cited by:4 (Source: Crossref)

Abstract

We study parafermion vertex algebras $N_{- 3 / 2} (𝔰 𝔩 (2))$ and $N_{- 3 / 2} (𝔰 𝔩 (3))$ . Using the isomorphism between $N_{- 3 / 2} (𝔰 𝔩 (3))$ and the logarithmic vertex algebra $𝒲^{0} {(2)}_{A_{2}}$ from [D. Adamović, A realization of certain modules for the $N = 4$ superconformal algebra and the affine Lie algebra $A_{2}^{(1)}$ , Transform. Groups21(2) (2016) 299–327], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra $𝒲 (2, 3)$ of central charge $c = - 10$ , and that $N_{- 3 / 2} (𝔰 𝔩 (3))$ is a direct sum of irreducible $N_{- 3 / 2} (𝔰 𝔩 (2))$ -modules. As a byproduct, we prove certain conjectures about the vertex algebra $𝒲^{0} {(p)}_{A_{2}}$ . We also obtain a vertex-algebraic proof of the irreducibility of a family of $𝒲 {(2, 3)}_{c}$ modules at $c = - 10$ .

Keywords:

AMSC: 17B69, 17B20, 17B65

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