TY - JOUR
T1 - From solitons to many-body systems
AU - Ben-Zvi, David
AU - Nevins, Thomas
N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2008
Y1 - 2008
N2 - 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].
AB - 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].
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U2 - 10.4310/pamq.2008.v4.n2.a3
DO - 10.4310/pamq.2008.v4.n2.a3
M3 - Article
AN - SCOPUS:57649097735
SN - 1558-8599
VL - 4
SP - 319
EP - 361
JO - Pure and Applied Mathematics Quarterly
JF - Pure and Applied Mathematics Quarterly
IS - 2 PART 1
ER -