On Ruijsenaars-Schneider spectrum from superconformal indices and ramified instantons

We discuss two physics-inspired approaches to derivation of the eigenfunctions and eigenvalues of A_N Ruijsenaars-Schneider model. First approach which was recently proposed by the authors relies on the computations of superconformal indices of class S 4dN = 2 theories with the insertion of surface defects. Second approach uses computations of Nekrasov-Shatashvili limit of 5dN = 1^* instanton partition functions in the presence of co-dimension two defect. We compare results of these two approaches for the low-lying levels of Ruijsenaars-Schneider model. We also discuss different previously proposed exact quantization conditions for the Coulomb branch parameters of the instanton partition functions and their interpretations in terms of index calculations.