From solitons to many-body systems

We present a bridge between the KP soliton equations and the Calogero-Moser many-body systems through noncommutative algebraic geometry. The Calogero-Moser systems have a natural geometric interpretation as flows on spaces of spectral curves on a ruled surface. We explain how the meromorphic solutions of the KP hierarchy have an interpretation via a noncommutative ruled surface. Namely, we identify KP Lax operators with vector bundles on quantized cotangent spaces (formulated technically in terms of D-modules). A geometric duality (a variant of the Fourier-Mukai transform) then identifies the parameter space for such vector bundles with that for the spectral curves and sends the KP flows to the Calogero{Moser flows. It follows that the motion and collisions of the poles of the rational, trigonometric and elliptic solutions of the KP hierarchy, as well as of its multicomponent analogs, are governed by the (spin) Calogero{Moser systems on cuspidal, nodal and smooth genus one curves. This provides geometric explanations and generalizations of results of Airault-McKean-Moser, Krichever and Wilson. The present paper is an overview of work to appear in [BN2].