Completeness from gravitational scattering

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 non-Abelian symmetry <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>G</a:mi> </a:math> , 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>G</c:mi> <c:mo>=</c:mo> <c:mi>S</c:mi> <c:mi>O</c:mi> <c:mo stretchy="false">(</c:mo> <c:mi>N</c:mi> <c:mo stretchy="false">)</c:mo> </c:math> with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>N</g:mi> <g:mo>≥</g:mo> <g:mn>5</g:mn> </g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>G</i:mi> <i:mo>=</i:mo> <i:mi>S</i:mi> <i:mi>U</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>N</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> , which in turn implies completeness for other symmetry groups such as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mrow> <o:mi>S</o:mi> <o:mi>p</o:mi> <o:mi>i</o:mi> <o:mi>n</o:mi> <o:mo stretchy="false">(</o:mo> <o:mi>N</o:mi> <o:mo stretchy="false">)</o:mo> </o:mrow> </o:math> , <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>S</s:mi> <s:mi>p</s:mi> <s:mo stretchy="false">(</s:mo> <s:mi>N</s:mi> <s:mo stretchy="false">)</s:mo> </s:math> , and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:msub> <w:mi>E</w:mi> <w:mn>8</w:mn> </w:msub> </w:math> . Curiously, a corollary of our results is that the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>S</y:mi> <y:mi>U</y:mi> <y:mo stretchy="false">(</y:mo> <y:mn>5</y:mn> <y:mo stretchy="false">)</y:mo> </y:math> and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>S</cb:mi> <cb:mi>O</cb:mi> <cb:mo stretchy="false">(</cb:mo> <cb:mn>10</cb:mn> <cb:mo stretchy="false">)</cb:mo> </cb:math> grand unified theories have precisely the minimal field content needed to derive completeness using our methodology.