On non-weight representations of the untwisted $N=2$ superconformal algebras

In this paper, we construct a family of non-weight modules over the untwisted $N=2$ superconformal algebras. Those modules when regarded as modules over the Cartan subalgebra are free of rank $2$. We give a classification of isomorphism classes of such modules. Moreover, all submodules of such modules are precisely determined. In particular, those modules we construct are not simple.