On non-relativistic integrable models and 4d SCFTs

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.