The Characteristic Dimension of Four-dimensional $\mathcal{N} = 2$ SCFTs

TL;DR

This paper introduces the characteristic dimension as a simple invariant for 4d $ =2$ SCFTs, revealing only nine possible values and constraining the geometry and spectra of these theories, while predicting new $ =3$ theories.

Contribution

It defines the characteristic dimension for 4d $ =2$ SCFTs, classifies its possible values, and links these to geometric and spectral properties, also predicting new $ =3$ theories.

Findings

Nine allowed values of the characteristic dimension.

Constraints on special K"ahler geometries for non-trivial dimensions.

Prediction of new $ =3$ theories with non-trivial one-form symmetries.

Abstract

In this paper we introduce the characteristic dimension of a four dimensional $\mathcal{N}=2$ superconformal field theory, which is an extraordinary simple invariant determined by the scaling dimensions of its Coulomb branch operators. We prove that only nine values of the characteristic dimension are allowed, $-\infty$, 1 ,6/5, 4/3, 3/2, 2, 3, 4, and 6, thus giving a new organizing principle to the vast landscape of 4d $\mathcal N=2$ SCFTs. Whenever the characteristic dimension differs from 1 or 2, only very constrained special K\"ahler geometries (i.e. isotrivial, diagonal and rigid) are compatible with the corresponding set of Coulomb branch dimensions and extremely special, maximally strongly coupled, BPS spectra are allowed for the theories which realize them. Our discussion applies to superconformal field theories of arbitrary rank, i.e. with Coulomb branches of any complex dimension. Along the way, we predict the existence of new $\mathcal{N}=3$ theories of rank two with non-trivial one-form symmetries.