S-duality and the N=2 Lens Space Index

TL;DR

This paper explores the mathematical structure of lens space indices in 4d N=2 theories, revealing their connection to S-duality, difference operators, and integrable models like the elliptic Ruijsenaars-Schneider system.

Contribution

It identifies the difference operators acting on lens space indices as a matrix-valued extension of the elliptic Ruijsenaars-Schneider model, linking physical dualities to integrable systems.

Findings

Lens space indices are constrained by S-duality.

Difference operators form a matrix-valued elliptic Ruijsenaars-Schneider model.

Eigenfunctions relate to non-symmetric Macdonald polynomials.

Abstract

We discuss some of the analytic properties of lens space indices for 4d N=2 theories of class S. The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects. We explicitly identify the difference operators to be a matrix-valued generalization of the elliptic Ruijsenaars-Schneider model. In a special limit these difference operators can be expressed naturally in terms of Cherednik operators appearing in the double affine Hecke algebras, with the eigenfunctions given by non-symmetric Macdonald polynomials.