Rationality in four dimensions

By leveraging the physics of the Higgs branch, we argue that the conformal central charges <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>a</a:mi></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>c</c:mi></c:math> of an arbitrary 4D <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi mathvariant="script">N</e:mi><e:mo>=</e:mo><e:mn>2</e:mn></e:math> superconformal field theory (SCFT) are rational numbers. Our proof of the rationality of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>c</h:mi></h:math> is conditioned on a well-supported conjecture about how the Higgs branch of an SCFT is encoded in its protected chiral algebra. To establish the rationality of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>a</j:mi></j:math>, we further rely on a widely believed technical assumption on the high-temperature limit of the superconformal index.