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$N=2$ super $W$ algebra in half-twisted Landau-Ginzburg model

We investigate $N=2$ extended superconformal symmetry, using the half-twisted Landau-Ginzburg models. The first example is the $D_{2n+2}$ -type minimal model. It has been conjectured that this model has a spin $n$ super $W$ current. We checked this by the direct computations of the BRS cohomology class up to $n=4$. We observe for $n\le 3$ the super W currents generate the ring isomorphic to the chiral ring of the model with respect to the classical product. We thus conjecture that this isomorphism holds for any $n$. The next example is $ CP_{n}$ coset model. In this case we find a sort of Miura transformation which gives the simple formula for the super W currents of spin \{1,2,...,n\} in terms of the chiral superfields. Explicit form of the super W currents and their Poisson brackets are obtained for $CP_{2},CP_{3}$ case. We also conjecture that as long as the classical product is concerned, these super W currents generate the ring isomorphic to the chiral ring of the model and this is checked for $CP_2$ model.