A realization of certain modules for the $N=4$ superconformal algebra and the affine Lie algebra $A_2^{(1)}$

Abstract

We shall first present an explicit realization of the simple $N=4$ superconformal vertex algebra $L_c^{N=4}$ with central charge $c=-9$. This vertex superalgebra is realized inside of the $bc\beta \gamma$ system and contains a subalgebra isomorphic to the simple affine vertex algebra $L_{A_1}(-\frac{3}{2}{\Lambda }_0)$. Then we construct a functor from the category of $L_c^{N=4}$--modules with $c=-9$ to the category of modules for the admissible affine vertex algebra $L_{A_2}(-\frac{3}{2}{\Lambda }_0)$. By using this construction we construct a family of weight and logarithmic modules for $L_c^{N=4}$ and $L_{A_2}(-\frac{3}{2}{\Lambda }_0)$. We also show that a coset subalgebra of $L_{A_2}(-\frac{3}{2}{\Lambda }_0)$ is an logarithmic extension of the $W(2,3)$--algebra with $c=-10$. We discuss some generalizations of our construction based on the extension of affine vertex algebra $L_{A_1}(k{\Lambda }_0)$ such that $k+2=1/p$ and $p$ is a positive integer.

arXiv comment27 pages

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Cited by in corpus (5)

• VOAs labelled by complex reflection groups and 4d SCFTs
• Simple current extensions beyond semi-simplicity
• Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results
• 4d $\mathcal{N}=2$ SCFTs and lisse W-algebras
• Staggered modules of $N=2$ superconformal minimal models