$\mathtt M$-曲线上的实正则KP因子与完全非负Grassmannian
Real regular KP divisors on $\mathtt M$-curves and totally non-negative Grassmannians
摘要：
在本文中，我们从表示完全非负Grassmannian $Gr^{\mbox{TNN}}(k,n)$中给定的不可约正向胞元$S$的平面双色（plabic）三价图到相关类别的实正则Kadomtsev-Petviashvili II（KP）解的谱数据构建了一个显式映射，从而完成了[4，6]中开始的KP方程的实代数几何数据的搜索。该谱曲线是退化有限间隙解的Krichever构造的，是一个有理退化的$M$-曲线，$Γ$，与图对偶。这些因子是$Γ$椭圆中的实正则KP因子，即它们满足[25]中选择实正则有限间隙解KPII解的条件。由于孤子数据是由$S$中的点描述的，因此我们在真正的正则有限隙KP解[25]和真正的正则多线KP孤子之间建立了一个桥梁，这些孤子已知由$Gr^{\mbox{TNN}}(k,n)$ [18，43]中的点参数化。我们利用文献[7]中提出的平面网络上关系空间的几何特征证明了这种构造对于网络上的许多规范自由度的不变性。这种关系系统在[53]中提出，用于计算在壳图$N=4$ SYM \cite{AGP1}上的散射振幅，并支配小的正格拉斯曼图$Gr^{\mbox{TP}}(1,3)$和$Gr^{\mbox{TP}}(2,3)$到任何给定的正向胞元$S$的完全非负融合。在我们的设置中，他们统治的现实性和规律性的KP因子。最后，我们解释了在Postnikov移动和约化下以及在正向细胞合并下曲线和除数的变换，并将我们的构造应用于一些例子。
英文摘要：
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.
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