论文标题
$ \ mathtt m $ curves和完全非负grassmannians上的真正常规KP除数
Real regular KP divisors on $\mathtt M$-curves and totally non-negative Grassmannians
论文作者
论文摘要
在本文中，我们在完全非负grassmannian $ gr^{\ mbox {\ mbox {tnn}}（tnn}}（k，n）中，代表给定的不可还能的正常grassmannian $ gr^{k，kp k. KP方程的真实代数几何数据的[4,6]始于[4,6]。该光谱曲线是基于简并有限差距解决方案的krichever构造建模的，并且是一个合理退化的$ m $ curve，$γ$，dual to Graph。除数是$γ$的椭圆形中的真正常规KP除数，即它们满足[25]中选择实际常规有限溶液KPII解决方案的条件。由于$ s $中的点描述了孤子数据，因此我们在实际的常规有限宽度KP解决方案[25]和真实的常规多行KP Solitons之间建立了一个桥梁，这些桥梁已知被$ gr^{\ mbox {tnn}}（k，k，n）$ [18,43]中的点进行参数化。 我们使用[7]中引入的Plabic网络上关系空间的几何表征，以证明这种构建相对于网络上许多量规自由的不变性。在[53]中提出了这种关系系统，以计算在壳图上的散射幅度$ n = 4 $ sym \ cit \ cite {agp1}，并控制着完全非阴性的小果态良好的良grass虫， $ gr^{\ mbox {tp}}（2,3）$，纳入任何给定的pocitroid cell $ s $。在我们的环境中，他们统治了KP除数的现实和规律性。 最后，我们解释了曲线和除数的转化，既然是在尼科夫的移动和减少和阳性细胞合并下的下降，并将我们的构造应用于一些例子。
In this paper we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell $S$ in the totally non-negative Grassmannian $Gr^{\mbox{TNN}}(k,n)$ to the spectral data for the relevant class of real regular Kadomtsev-Petviashvili II (KP) solutions, thus completing search of real algebraic-geometric data for the KP equation started in [4,6]. The spectral curve is modeled on Krichever construction for degenerate finite-gap solutions, and is a rationally degenerate $M$-curve, $Γ$, dual to the graph. The divisors are real regular KP divisors in the ovals of $Γ$, i.e. they fulfill the conditions for selecting real regular finite--gap solutions KPII solutions in [25]. Since the soliton data are described by points in $S$, we establish a bridge between real regular finite-gap KP solutions [25] and real regular multi-line KP solitons which are known to be parameterized by points in $Gr^{\mbox{TNN}}(k,n)$ [18,43]. We use the geometric characterization of spaces of relations on plabic networks introduced in [7] to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in [53] for the computation of scattering amplitudes on on--shell diagrams $N=4$ SYM \cite{AGP1} and govern the totally non--negative amalgamation of the little positive Grassmannians, $Gr^{\mbox{TP}}(1,3)$ and $Gr^{\mbox{TP}}(2,3)$, into any given positroid cell $S$. In our setting they rule the reality and regularity properties of the KP divisor. Finally, we explain the transformation of the curve and the divisor both under Postnikov moves and reductions and under amalgamation of positroid cells, and apply our construction to some examples.
