On Ruijsenaars-Schneider spectrum from superconformal indices and ramified instantons

TL;DR

This paper explores two physics-inspired methods to derive eigenfunctions and eigenvalues of the Ruijsenaars-Schneider model, connecting superconformal indices, instanton partition functions, and defect insertions.

Contribution

It introduces and compares two novel approaches based on superconformal indices and instanton partition functions for solving the Ruijsenaars-Schneider model.

Findings

Results agree for low-lying energy levels

Provides insights into quantization conditions for Coulomb branch parameters

Links between index calculations and eigenvalue problems

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 $5d$ ${\mathcal N} = 1^*$ 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.