Completeness from Gravitational Scattering

TL;DR

This paper proves that in gravitational theories with certain symmetries, the existence of finitely many charged particles implies an infinite charge lattice, supporting the completeness hypothesis.

Contribution

It demonstrates that symmetry and gravitational consistency enforce the necessity of infinitely many charged particles in a broad class of theories.

Findings

Finitely many charged particles imply an infinite charge lattice.

Completeness is shown for groups like SO(N), SU(N), Spin(N), Sp(N), and E8.

Grand unified theories SU(5) and SO(10) have minimal content for completeness.

Abstract

We prove that symmetry in the presence of gravity implies a version of the completeness hypothesis. For a broad class of theories, we demonstrate that the existence of finitely many charged particles logically necessitates the existence of infinitely many charged particles populating the entire charge lattice. Our conclusions follow from the consistency of perturbative gravitational scattering and require the following ingredients: 1) a weakly coupled ultraviolet completion of gravity, 2) a nonabelian symmetry $G$, gauged or global, whose Cartan subgroup generates the abelian charge lattice, and 3) a spectrum containing some finite set of charged representations, in the simplest cases taken to be a single particle in the fundamental. Under these conditions, the abelian charge lattice is completely filled by single-particle states for $G=SO(N)$ with $N\geq 5$ and $G=SU(N)$ with $N\geq 3$, which in turn implies completeness for other symmetry groups such as $Spin(N)$, $Sp(N)$, and $E_8$. Curiously, a corollary of our results is that the $SU(5)$ and $SO(10)$ grand unified theories have precisely the minimal field content needed to derive completeness using our methodology.