$\mathcal{N}=1$ Supersymmetric Double Field Theory and the generalized Kerr-Schild Ansatz

We construct the $\mathcal{N} = 1$ supersymmetric extension of the generalized Kerr-Schild ansatz in the flux formulation of Double Field Theory. We show that this ansatz is compatible with $\mathcal{N} = 1$ supersymmetry as long as it is not written in terms of generalized null vectors. Supersymmetric consistency is obtained through a set of conditions that imply linearity of the generalized gravitino perturbation and unrestricted perturbations of the generalized background dilaton and dilatino. As a final step we parametrize the previous theory in terms of the field content of the low energy effective $10$-dimensional heterotic supergravity and we find that the perturbation of the $10$-dimensional vielbein, Kalb-Ramond field and gravitino can be written in terms of a pair of null vectors, as expected.