(Super)$\,$Gravity from Positivity

We investigate whether the effective theory for isolated, massive, and weakly interacting spin-$3/2$ particles is compatible with causality and unitarity-i.e., the positivity of scattering amplitudes. We find no solution to positivity constraints, except when gravitons are also present and couple in a (nearly) supersymmetric way. Gravity is thus bootstrapped from $S$-matrix consistency conditions for the longitudinal and transverse polarizations of massive spin-$3/2$ states. For two such particles forming a $U(1)$-charged state, a (gravi)photon gauging the symmetry is also required, with couplings characteristic of supergravity and consistent with both the no global symmetry and weak gravity conjectures. We further explore the EFT-hedron associated with the longitudinal polarizations, the Goldstinos, through novel $t$-$u$ symmetric dispersion relations. We identify the extremal UV models that lie at the corners of the allowed parameter space, recovering familiar models of supersymmetry breaking and uncovering new ones.