Bosonization and Fermionization of the Superstring Oscillators

In this manuscript we consider the transformations of the oscillators of the bosonic fields of the superstring in terms of the fermions oscillators and vice versa. We demand the exchange of the commutation and anti-commutation relations of the oscillators. Therefore, we obtain some conditions on the Grassmannian matrices that appear in these transformations. We observe that there are several methods to obtain these conditions. In addition, adjoints of the matrix elements and $T$-duality of these matrices will be obtained. The effects of this bosonization and fermionization on the mass operators and on some massless states will be studied. The covariant formalism will be used and hence we consider both the matter parts and the ghost parts of the superstring theory.