Extremal unitary representations of big $N=4$ superconformal algebra

In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big $N=4$ superconformal algebra which can be found in [5]. Our results agree with the general conjectures about classification of unitary highest weight representation of minimal $W$-algebras attached to basic Lie superalgebras formulated in [10], [11], and complete their proof for the big $N=4$ superconformal algebra.