Large N universality of 4d \( \mathcal{N} \) = 1 superconformal index and AdS black holes

We study the large N limit of the matrix models associated with the superconformal indices of four-dimensional $\mathcal{N}$ \( \mathcal{N} \) = 1 superconformal field theories. We find that for a large class of $\mathcal{N}$ \( \mathcal{N} \) = 1 superconformal gauge theories, the superconformal indices in the large N limit of such theories are dominated by the ‘parallelogram’ saddle, providing O(N²) free energy for the generic value of chemical potentials. This saddle corresponds to BPS black holes in AdS₅ whenever a holographic dual description is available. Our saddle applies to a large class of gauge theories, including ADE quiver gauge theories, and the theories with rank-2 tensor matters. Our analysis works for most $\mathcal{N}$ \( \mathcal{N} \) = 1 superconformal gauge theories that admit a suitable large N limit while keeping the flavor symmetry fixed. We also find ‘multi-cut’ saddle points, which correspond to the orbifolded Euclidean black holes in AdS₅.