Infinitely many 4d $\mathcal{N}=2$ SCFTs with $a=c$ and beyond

TL;DR

This paper introduces a new class of 4d $ abla=2$ SCFTs labeled by Lie groups, revealing infinite examples with equal central charges and expressing their indices in terms of well-known mathematical functions, expanding understanding of non-Lagrangian theories.

Contribution

It constructs a broad family of non-Lagrangian 4d $ abla=2$ SCFTs with $a=c$, providing explicit index formulas and connecting to mathematical functions like MacMahon's sum-of-divisors, extending the landscape of known theories.

Findings

Existence of infinitely many 4d $ abla=2$ SCFTs with $a=c$

Schur indices expressed via $ abla=4$ super Yang--Mills indices and special functions

Connection to affine quiver gauge theories and Deligne--Cvitanovi\'c series

Abstract

We study a set of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) $\widehat{\Gamma}(G)$ labeled by a pair of simply-laced Lie groups $\Gamma$ and $G$. They are constructed out of gauging a number of $\mathcal{D}_p(G)$ and $(G, G)$ conformal matter SCFTs; therefore they do not have Lagrangian descriptions in general. For $\Gamma = D_4, E_6, E_7, E_8$ and some special choices of $G$, the resulting theories have identical central charges $(a=c)$ without taking any large $N$ limit. Moreover, we find that the Schur indices for such theories can be written in terms of that of $\mathcal{N}=4$ super Yang--Mills theory upon rescaling fugacities. Especially, we find that the Schur index of $\widehat{D}_4(SU(N))$ theory for $N$ odd is written in terms of MacMahon's generalized sum-of-divisor function, which is quasi-modular. For generic choices of $\Gamma$ and $G$, it can be regarded as a generalization of the affine quiver gauge theory obtained from $D3$-branes probing an ALE singularity of type $\Gamma$. We also comment on a tantalizing connection regarding the theories labeled by $\Gamma$ in the Deligne--Cvitanovi\'c exceptional series.