With T. Nevins. Toda lattice hierarchy and noncommutative geometry.
We apply the algebraic geometry of additive, multiplicative and elliptic difference operators using generalizations of the Fourier-Mukai transform. Namely we construct moduli spaces of difference modules on (cuspidal, nodal or smooth) cubic curves and identify them with moduli spaces of spectral sheaves on C^* bundles over the same curves. We then apply this description to a problem of integrable systems explored by Krichever and Zabrodin: the relation between rational, trigonometric and elliptic solutions of the Toda lattice hierarchy (and its nonabelian generalizations) and the Ruijsenaars-Schneiders (RS) particle systems (and their spin variants). We show that the Toda lattice has a natural interpretation as flows on difference modules and that the Fourier-Mukai transform puts the Toda hierarchy in action-angle form, identifying the difference modules with the spectral curves of the RS systems and the poles of the Toda solutions with the positions of the RS particles.