Supersymmetric semi--infinite cohomology and $N=2$ superconformal structures

We study supersymmetric semi-infinite cohomology and its $N=2$ superconformal structure via the BRST reduction of an $N=\frac{3}{2}$ superconformal structure on the super-affine vertex algebra associated to a super Manin pair arising from the Kazama–Suzuki $N=2$ condition. Moreover, we realize super coset models in terms of relative supersymmetric semi-infinite cohomology.