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arXiv · 2005.02631

On parafermion vertex algebras of $\frak{sl}(2)_{-3/2}$ and $\frak{sl}(3)_{-3/2}$

Drazen Adamovic·Antun Milas·Qing Wang
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Abstract

We study parafermion vertex algebras $N_{-3/2}(\frak{sl}(2))$ and $N_{-3/2}(\frak{sl}(3))$. Using the isomorphism between $N_{-3/2}(\frak{sl}(3))$ and the logarithmic vertex algebra $\mathcal{W}^{0} (2)_{A_2} $ from [2], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra $\mathcal{W}(2,3)$ of central charge $c=-10$, and that $N_{-3/2}(\frak{sl}(3))$ is a direct sum of irreducible $N_{-3/2}(\frak{sl}(2))$-modules. As a byproduct, we prove certain conjectures about the vertex algebra $\mathcal{W}^0(p)_{A_2}$. We also obtain a vertex-algebraic proof of the irreducibility of a family of $\mathcal W(2,3)_{c}$ modules at $c=-10$.

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Drazen Adamovic, Antun Milas, Qing Wang. 2020-05-06. On parafermion vertex algebras of $\frak{sl}(2)_{-3/2}$ and $\frak{sl}(3)_{-3/2}$. https://arxiv.org/abs/2005.02631

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