We utilize dispersion relations to constrain weakly coupled ultraviolet (UV) completions of gravity when the external particles are scalars with color or flavor indices. We furnish a sum rule which implies that only certain patterns of representations can be ommitted in the UV. For adjoint scalars coupled to gravity, this predicts new particles, or the breaking of the symmetry, in the UV. This is all derived from the boundedness required for the dispersion relation, covariance with the underlying symmetries, and crossing symmetry. Unitarity is not invoked to derive the vanishing constraint.

Extending previous work on a dual resonant bootstrap which has the Veneziano amplitude as its unique solution, we motivate a closed string analogue of the bootstrap which inspires novel deformations of the Virasoro-Shapiro amplitude. This amplitude interpolates between the corners of the EFT-hedron defined by Caron-Huot and Van Duong and sweeps out EFT coefficient trajectories beyond the trivial higher-spin region.

We motivate an analytic bootstrap which defines algebraic varities whose solutions are multi-parameter families of dual resonant tree-level amplitudes. Imposing the strongest form of our assumptions leaves only a single point, the Veneziano amplitude. This defines simple conditions under which the Veneziano amplitude is the unique consistent solution to a bootstrap problem.