Towards the classification of rank-$r$ $\mathcal{N}=2$ SCFTs. Part I: twisted partition function and central charge formulae

TL;DR

The work establishes a general framework to compute the conformal central charges $a$ and $c$ of four-dimensional ${\mathcal N}=2$ SCFTs from Coulomb branch data at arbitrary rank by connecting the twisted partition function to CB singularities. It introduces discriminant concepts (discriminant, quantum discriminant, physical discriminant) and demonstrates how rank-1 theories living on CB strata determine higher-rank data via explicit formulae. A key result is the ${\mathcal N}=2$ UV-IR simple flavor condition, which reduces mass deformations of higher-rank theories to deformations of rank-1 SCFTs, thereby unifying the analysis across ranks. The central charges are expressed in terms of rank-1 data through concrete formulae that depend on CB dimensions, ECB quaternionic dimension, and stratified IR theories, with consistency checks in explicit SU(3) rank-2 examples and compatibility with known bounds and S-dual relations.

Abstract

We derive explicit formulae to compute the $a$ and $c$ central charges of four dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the low-energy limit behavior of the flavor symmetry of $\mathcal{N}=2$ SCFTs which culminate with our $\mathcal{N}=2$ UV-IR simple flavor condition. This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d $\mathcal{N}=2$ SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank-2 SCFTs, including new ones, in a companion paper.