Stringy Completions of the Standard Model from the Bottom Up

It is notoriously difficult to construct UV completions of gravitational amplitudes, in particular those of the Standard Model coupled to gravity. On the other hand, constraints on EFT coefficients from unitarity, locality, and causality alone do not constraint the spaces commensurately, in particular string amplitudes occupy special places in these allowed regions, but certainly do not span them.  In order to explore the possibility of interesting constraints on a class of Ansätze for gravitational completions, we pick a half-way place between explicit string compactifications and totally general dispersive analyses, which apply directly to amplitudes in the Standard Model and putative GUT models.  We find interesting constraints relating the gauge coupling, the string scale in Planck units, the rank of the gauge group, and the space-time dimension, in these models.  These results further motivate the effort to understand how the particularly constrained form of stringy gravitational completions is manifest analytically and could be exploited in more general settings.