Minimal unitary representations from supersymmetry

We compute the supersymmetry constraints on the R4 type corrections in maximal supergravity in dimension 8, 6, 4 and 3, and determine the tensorial differential equations satisfied by the function of the scalar fields multiplying the R4 term in the corresponding invariants. The second-order derivative of this function restricted to the Joseph ideal vanishes in dimension lower than six. These results are extended to the ∇4R4 and the ∇6R4 corrections, based on the harmonic superspace construction of these invariants in the lin earised approximation. We discuss the solutions of these differential equations and analysis the consequences on the non-perturbative type II low energy string theory effective action.