Infinitely many 4D N = 2 SCFTs with a = c and beyond

We study a set of four-dimensional N=2 superconformal field theories (SCFTs) ^Γ(G) labeled by a pair of simply laced Lie groups Γ and G. They are constructed out of gauging a number of D_p(G) and (G,G) conformal matter SCFTs; therefore, they do not have Lagrangian descriptions in general. For Γ=D₄,E₆,E₇,E₈, 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 N=4 super-Yang-Mills theory upon rescaling fugacities. Especially, we find that the Schur index of ^D₄(SU(N)) theory for N odd is written in terms of MacMahon's generalized sum-of-divisor function, which is quasimodular. For generic choices of Γ and G, it can be regarded as a generalization of the affine quiver gauge theory obtained from D3-branes probing singularity of type Γ. We also comment on a tantalizing connection regarding the theories labeled by Γ in the Deligne-Cvitanović exceptional series.