Rationality in Four Dimensions

TL;DR

This paper argues that the conformal central charges of 4d N=2 superconformal field theories are rational numbers, based on physics of the Higgs branch and certain conjectures, advancing understanding of SCFT invariants.

Contribution

It provides a proof of the rationality of the central charge c and a, relying on new physical insights and conjectures about the Higgs branch and superconformal index.

Findings

Central charges a and c are rational numbers for 4d N=2 SCFTs.

Proof of c's rationality depends on a conjecture about the Higgs branch and chiral algebra.

Rationality of a relies on a high-temperature limit assumption of the superconformal index.

Abstract

By leveraging the physics of the Higgs branch, we argue that the conformal central charges $a$ and $c$ of an arbitrary 4d $N=2$ superconformal field theory (SCFT) are rational numbers. Our proof of the rationality of $c$ 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 $a$, we further rely on a widely-believed technical assumption on the high-temperature limit of the superconformal index.