Non-linear structures in the non-critical NSR string

We investigate the Ward identities of theW symmetry in the super-Liouville theory coupled to the super-conformal matter of central charge . The theory is classified into two chiralities. For the positive chirality, all gravitationally dressed scaling operators are generated from theq–1 gravitational primaries by one of the ring generators in the R-sector acting on them repeatedly. After fixing the normalizations of the dressed scaling operators, we find that the Ward identities are expressed in the form of theusual W _q algebra constraints as in the bosonic case:W _n ^(k+1) ,=0, (k=1,...,q–1;nZ1–k), where the equations for even and oddn come from the currents in the NS- and the R-sector respectively. The non-linear terms come from the anomalous contributions at the boundaries of moduli space. The negative chirality is defined by interchanging the roles ofp andq. Then we get theW _p algebra constraints.