Nambu-Dirac manifolds

We present a generalization of the Dirac structure in the direction of the Nambu-Poisson structure. It is shown that with every integrable Nambu-Dirac structure there is associated a Leibniz algebroid, which yields a singular foliation endowed with a closed form. The examples of Nambu-Dirac manifolds are Dirac manifolds, Nambu-Poisson manifolds and manifolds with closed forms. A different Leibniz algebroid structure associated with a Nambu-Poisson structure is adopted for this generalization.