We shall first present an explicit realization of the simple N=4
superconformal vertex algebra LcN=4β with central charge c=β9. This
vertex superalgebra is realized inside of the bcΞ²Ξ³ system and
contains a subalgebra isomorphic to the simple affine vertex algebra LA1ββ(β23βΞ0β). Then we construct a functor from the category of
LcN=4β--modules with c=β9 to the category of modules for the
admissible affine vertex algebra LA2ββ(β23βΞ0β). By
using this construction we construct a family of weight and logarithmic modules
for LcN=4β and LA2ββ(β23βΞ0β). We also show
that a coset subalgebra of LA2ββ(β23βΞ0β) is an
logarithmic extension of the W(2,3)--algebra with c=β10. We discuss some
generalizations of our construction based on the extension of affine vertex
algebra LA1ββ(kΞ0β) such that k+2=1/p and p is a positive
integer.Comment: 27 page