[Paper Review] Real regular KP divisors on $\mathtt M$-curves and totally non-negative Grassmannians

Simonetta Abenda, P. G. Grinevich|arXiv (Cornell University)|Feb 12, 2020

Algebraic structures and combinatorial models60 references4 citations

TL;DR

This paper constructs an explicit map from planar bicolored trivalent graphs (plabic networks) representing irreducible positroid cells in the totally non-negative Grassmannian $\mathrm{Gr}^{\mathrm{TNN}}(k,n)$ to real regular KP divisors on $\mathtt{M}$-curves, completing a geometric bridge between finite-gap solutions and multi-line soliton solutions of the KP equation. The construction uses spectral curves dual to the graphs and establishes invariance under Postnikov moves and reductions via systems of relations on networks, with divisor regularity governed by totally non-negative amalgamation structures.

ABSTRACT

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.

Motivation & Objective

• To complete the geometric correspondence between real regular finite-gap KP solutions and real regular multi-line soliton solutions.
• To establish a map from irreducible positroid cells in $\mathrm{Gr}^{\mathrm{TNN}}(k,n)$ to real regular KP divisors on $\mathtt{M}$-curves.
• To prove invariance of the divisor construction under gauge freedoms and Postnikov moves using systems of relations on plabic networks.
• To characterize divisor regularity and behavior under amalgamation and curve desingularization.
• To resolve singularities in degenerate cases where wave functions vanish at nodes, using gauge freedom and limit constructions.

Proposed method

• Constructs a rationally degenerate $\mathtt{M}$-curve $\Gamma$ dual to a trivalent plabic graph representing a positroid cell in $\mathrm{Gr}^{\mathrm{TNN}}(k,n)$.
• Defines a degree-$g$ divisor on $\Gamma$ using the Baker–Akhiezer function and pole structure of the KP wave function.
• Uses systems of relations on plabic networks—previously introduced in [7]—to ensure invariance under gauge transformations and Postnikov moves.
• Applies the Sato divisor and local coordinates to characterize divisor points on ovals and components of the curve.
• Employs gauge freedom on unreduced graphs to eliminate identically zero wave functions and avoid singularities.
• Analyzes divisor behavior under amalgamation of positroid cells and curve transformations via desingularization and limit processes.

Experimental results

Research questions

• RQ1 How can real regular KP divisors on $\mathtt{M}$-curves be explicitly constructed from soliton data in $\mathrm{Gr}^{\mathrm{TNN}}(k,n)$?
• RQ2 What is the role of systems of relations on plabic networks in ensuring invariance of the divisor construction under gauge freedom and Postnikov moves?
• RQ3 How do the divisor and spectral curve transform under amalgamation of positroid cells and reductions of the network?
• RQ4 What happens to the divisor when the wave function vanishes identically at edges or nodes, and how can such singularities be resolved?
• RQ5 Can the divisor construction be extended to reducible or degenerate plabic networks using limit processes and gauge choices?

Key findings

• The construction provides a complete, explicit map from irreducible positroid cells in $\mathrm{Gr}^{\mathrm{TNN}}(k,n)$ to real regular KP divisors on $\mathtt{M}$-curves dual to their plabic graphs.
• The divisor is invariant under Postnikov moves and reductions due to the underlying system of relations on the network, which governs reality and regularity.
• For reducible graphs, the divisor depends on edge weights unless gauge freedom is used to eliminate zero wave functions, as demonstrated in the $Gr^{\mathrm{TP}}(1,2)$ example.
• When the wave function vanishes identically at edges (e.g., at $a=1$), the divisor points coincide at double points, requiring a resolution of singularities similar to the irreducible case.
• In cases with identically zero wave functions, the divisor structure depends on parameter choices; however, gauge freedom can eliminate such degeneracies in unreduced graphs.
• The Sato divisor point $\gamma_S$ is independent of KP times and lies at the intersection of the oval $\Omega_1$ and the component $\Gamma_0$, while other divisor points depend on weights and network structure.