In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp–Lieb’s type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp–Lieb’s inequality and its generalization to compact Riemannian manifolds without boundary.