sℓ(2)−4 WZW MODEL AS AN (N=4)-SUPERSYMMETRIC BOSONIC STRING WITH c=−2 MATTER

We consider the sℓ(2) current algebra at level k=−4 when the sℓ(2) BRST operator is nilpotent. We formulate a spectral sequence converging to the cohomology of this BRST operator. At the second term in the spectral sequence, we observe the existence of an N=4 algebra. This algebra is generated in a c=−2 bosonic string whose BRST operator [Formula: see text] represents the next term in the spectral sequence. We realize the cohomology of the irreducible modules as [Formula: see text] primitives of theN=4 singular vectors and relate the latter to the Lian–Zuckerman states of c=−2 matter. The relation between the sℓ(2)−4 WZW model and the c=−2 bosonic string is established both at the level of BRST cohomology and at the level of an explicit operator construction. The relation of the N=4 algebra to the known symmetries of matter+gravity systems is also investigated.