Complete Graphs, Hilbert Series, and the Higgs branch of the 4d N=2 $(A_n,A_m)$ SCFT's

Understand

The strongly interacting 4d N=2 SCFT's of type $(A_n,A_m)$ are the simplest examples of models in the $(G,G^\prime)$ class introduced by Cecotti, Neitzke, and Vafa in arXiv:1006.3435.

• These systems have a known 3d N=4 mirror only if $h(A_n)$ divides $h(A_m)$, where $h$ is the Coxeter number.
• By 4d/2d correspondence, we show that in this case these systems have a nontrivial global flavor symmetry group and, therefore, a non-trivial Higgs branch.
• As an application of the methods of arXiv:1309.2657, we then compute the refined Hilbert series of the Coulomb branch of the 3d mirror for the simplest models in the series.