Completeness from Gravitational Scattering

TL;DR

This work derives a gravity-driven version of the completeness hypothesis by exploiting the dispersion relation for gravitational scattering, $c_2(t)=-\frac{8\pi G}{t}$ with $b_2(t)=0$, in theories with weakly coupled UV gravity and a nonabelian symmetry $G$ whose Cartan subgroup generates the abelian charge lattice. Starting from a finite seed spectrum, the authors iteratively scatter charges and orbit results under Weyl/outer automorphisms to populate the entire Cartan lattice, proving charge completeness for $SO(N)$ with $N\ge5$ and $SU(N)$ with $N\ge3$ and extending to $Spin(N)$, $Sp(N)$, and $E_8$. They show that for $SU(5)$ and $SO(10)$ GUTs, the minimal field content suffices to guarantee completeness, linking phenomenology to spectral completeness. The results rely on tree-level UV gravity, but point to broader implications for swampland ideas, higher-point bootstrap, and the role of nonabelian symmetries in constraining the spectrum of quantum gravity theories.

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.

Figures (4)

• Figure 1: The $SO(4)$ charge lattice, stratified according to the central charge sectors $z=0$ (black) and $z=1$ (gray). Overlaid is the sequence of scattering processes in Eq. (\ref{['eq:n_scattering']}). Starting from an initial spectrum composed of the fundamental (red), we scatter in succession (orange, yellow, green, blue, indigo) to obtain a set of ultracharged states (purple). We then apply lowering operators to generate all charges at the boundary and interior of the diamond (purple).
• Figure 2: The $SU(3)$ charge lattice, stratified according to the central charge sectors $z=0$ (black), $z=1$ (green), and $z=2$ (purple). The polygons circumscribe the irreducible representations $Q_{\mathbf{3}}$ (green), $Q_{ \bar{\mathbf{3}}}$ (purple), $Q_{ {\mathbf{8}}}$ (black), $Q_{ {\mathbf{10}}}$ (brown), and $Q_{ \overline{\mathbf{10}}}$ (gray).
• Figure 3: The $SU(3)$ charge lattice. Left: The seed of the iteration. Right: Sequence of polygons allowing iteration, namely, the dark blue triangle $T_n$, blue hexagon $H_n$, and light blue triangle $T_{n+3}$. Relevant points for scattering are denoted with circles.
• Figure 4: The $SU(3)$ charge lattice. Left: starting hexagon with the starting charge $P_0$ (dark blue). Right: sequence of hexagons (dark blue, blue, light blue), highlighting the relevant points $P_n$, (dark blue), $D_n$ (dark blue), and $P_{n+1}$ (blue).