Elliptic Ruijsenaars–Schneider and quantum K-theory spectral correspondence

Let $\mathscr E_r^{(nN)}$, for $r=1,\dots,nN$, denote the eigenvalues of the elliptic Ruijsenaars–Schneider Hamiltonians, and let $\mathscr U$ be the universal bundle over the moduli space $\mathcal M_N$ of rank-$N$ sheaves on the plane. Elliptic Ruijsenaars–Schneider and quantum K-theory conjecture. The eigenvalues of quantum multiplication by $\Lambda^r\mathscr U$ are in one-to-one correspondence with $\mathscr E_r^{(nN)}$. For $r=1$, this correspondence is given by

$$\mathscr E_1^{(N)}(\mathfrak z)=\lim_{n\to\infty}\left[\hbar^{n-1}(1-\hbar)\frac{(\mathfrak p\hbar;\mathfrak p)_\infty(\mathfrak p q\hbar;\mathfrak p)_\infty}{(\mathfrak p;\mathfrak p)_\infty(\mathfrak p q^{-1};\mathfrak p)_\infty}\mathscr E_1^{(nN)}\right],$$

with quantum parameter identified with the elliptic parameter by

$$\mathfrak z=-\mathfrak p\sqrt{q\hbar}.$$

This is the proposed spectral relationship between elliptic integrable systems and quantum K-theory of sheaf moduli spaces. The supplied text gives no resolution status.