A realization of certain modules for the n = 4 superconformal algebra and the affine lie algebra a 2 (1)

TL;DR: In this article, an explicit realization of the simple N = 4 superconformal vertex algebra L isEnabled A1 with central charge c = −9 is presented, which is realized inside of the bcβγ system and contains a subalgebra isomorphic to the simple affine vertex algebra.

Abstract: We shall first present an explicit realization of the simple N = 4 superconformal vertex algebra L  = 4 with central charge c = −9. This vertex superalgebra is realized inside of the bcβγ system and contains a subalgebra isomorphic to the simple affine vertex algebra L A1 $$ \left(-\frac{3}{2}{\varLambda}_0\right) $$ . Then we construct a functor from the category of L  = 4 -modules with c = −9 to the category of modules for the admissible affine vertex algebra L A1 $$ \left(-\frac{3}{2}{\varLambda}_0\right) $$ . By using this construction we construct a family of weight and logarithmic modules for L  = 4 and L A1 $$ \left(-\frac{3}{2}{\varLambda}_0\right) $$ . We also show that a coset subalgebra L A1 $$ \left(-\frac{3}{2}{\varLambda}_0\right) $$ is a 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 A1 (kΛ 0) such that k + 2 = 1/p and p is a positive integer.