Superconformal indices of N=4 dual field theories

Using the superconformal indices techniques, we construct new interesting conjectures for the elliptic hypergeometric integral identities based on the exceptional groups $G_2$ and $F_4$. These identities arise from $S$-dualities for $\mathcal{N}=4$ supersymmetric Yang-Mills theories with $G_2$ and $F_4$ gauge groups