从孤子到多体系统

摘要：

通过非对易代数几何的方法，我们在KP孤子方程和Calogero-Moser多体系统之间架起了一座桥梁。Calogero-Moser系统有一个自然的几何解释，即在直纹曲面上的谱曲线空间上流动。我们解释了如何亚纯解的KP层次有一个解释，通过一个非交换的直纹面。也就是说，我们确定KP Lax运营商的向量丛量化余切空间（制定技术上的D-模块）。然后，几何二元性（Fourier-Mukai变换的变体）将此类向量丛的参数空间与谱曲线的参数空间识别出来，并将KP流发送到Calogero-Moser流。由此可见，KP族的有理解、三角解和椭圆解的极点的运动和碰撞，以及其多分量类似物，都是由尖点、节点和光滑亏格1曲线上的（自旋）Calogero-Moser系统决定的。这为Airault-McKean-Moser、Krichever和Wilson的结果提供了几何解释和推广。

英文摘要：

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.

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