On N = 1 conformal manifolds of class S theories

We present an algorithm to search for Lagrangian SCFTs conformally dual to a given theory. We search the space of possible Lagrangian theories for conformal theories with desired properties: in particular number of vector fields, number of chiral fields, and the first two terms in the series expansion of the Schur limit of the superconformal index. We implement this algorithm to look for dual theories for some class S theories. Namely, for those theories with effective number of vector fields being 18 and 20. We find two possible new dualities relating gaugings of the R_{0,4} theory to quiver gauge theories. We also discuss a novel observation (examples of theories that have matching a, c anomalies, as well as the Schur limit of their index, but do not match in the full superconformal index). These relations invite more explorations.