On Ruijsenaars-Schneider spectrum from superconformal indices and ramified instantons

Abstract

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 $\mathcal{S}$ $4d$ $\mathcal{N}=2$ theories with the insertion of surface defects. Second approach uses computations of Nekrasov-Shatashvili limit of $5d$ $\mathcal{N}=1^{\ast }$ 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.

25 pages