With T. Nevins. **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 a geometric explanation
and generalizations of results of Airault-McKean-Moser, Krichever,
and Wilson.