On non-relativistic integrable models and 4d SCFTs

TL;DR

This paper explores the connection between 4d N=2 SCFT indices and non-relativistic integrable models, revealing new identities and relations to generalized Schur limits of N=1 theories.

Contribution

It explicitly relates Schur indices of class S theories to elliptic Jack functions and non-relativistic integrable models, uncovering new identities and mappings between different theories.

Findings

Schur indices expressed via elliptic Jack functions and Lamé eigenfunctions.

Identities linking indices of different theories in the Deligne-Cvitanović series.

Non-relativistic limits of integrable models relate to Schur-like limits of N=1 SCFTs.

Abstract

We elaborate on the relation between the generalized Schur index of $N=2$ SCFTs in four dimensions and the non-relativistic limit of the elliptic Ruijsenaars-Schneider model. In particular we discuss explicitly how to express generalized Schur indices of theories of class $S$ in terms of elliptic Jack functions. For example, in the $A_1$ case the indices are given naturally in terms of eigenfunctions of the Lam\'{e} equation. We use the expression in terms of eigenfunctions to further check the recent observation that the generalized Schur indices of different theories in the Deligne-Cvitanovi\'{c} series can be mapped onto each other. This mapping implies non trivial identities on unrefined sums of eigenfunctions of non-relativistic elliptic Calogero-Moser models associated to different root systems. We claim then that the non-relativistic limits of various integrable models give rise naturally to generalized Schur-like limits of classes of $N=1$ SCFTs. As an example we discuss the relation of the Inozemtsev model, the non relativistic limit of the van Diejen model, and compactifications of the rank $Q$ E-string theory. We argue that in general the ``Schur index'' of $N=1$ $4d$ SCFTs can be understood as being related to the free fermionic limit of a non-relativistic integrable model.