A comment on the enhancement of global symmetries in superconformal SU(2) gauge theories in 5D

Abstract

Recently, the superconformal index for a class of 5-dimensional superconformal quantum field theories with gauge group SU(2) and $N_f$ flavors was computed in Kim, Kim, and Lee (2012) to a few low orders in the chemical potential and enhancement of the global symmetry to $E_{N_f+1}$ was confirmed to this low order. In this note I provide an argument that the information contained in these few low order terms in the expansion of the index is sufficient to prove the enhancement of global symmetries in the theories. So, in particular the expression for the indices must display the enhancement to all orders in the chemical potential.