Non-linear structures in the non-critical NSR string

We investigate the Ward identities of the script W sign_∞ symmetry in the super-Liouville theory coupled to the super-conformal matter of central charge ĉ_M = 1 - 2(p - q)²/pq. The theory is classified into two chiralities. For the positive chirality, all gravitationally dressed scaling operators are generated from the q - 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 the usual script W sign algebra constraints as in the bosonic case: script W sign_n^(k+1)τ = 0, (k = 1, . . . , q - 1; n ∈ Z_≧1-k), where the equations for even and odd n 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 of p and q. Then we get the script W sign_p algebra constraints.