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

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.