Real regular KP divisors on $${\texttt {M}}$$-curves and totally non-negative Grassmannians

Abstract

In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}$$ S M TNN in the totally non-negative Grassmannian $$Gr^{\text{ TNN }}(k,n)$$ G r TNN ( k , n ) to the spectral data for the relevant class of real regular Kadomtsev–Petviashvili II (KP-II) solutions, thus completing the search of real algebraic-geometric data for the KP-II equation started in Abenda and Grinevich (Commun Math Phys 361(3):1029–1081, 2018; Sel Math New Ser 25(3):43, 2019). The spectral curve is modeled on the Krichever construction for degenerate finite-gap solutions and is a rationally degenerate $${\texttt {M}}$$ M -curve, $$\Gamma $$ Γ , dual to the graph. The divisors are real regular KP-II divisors in the ovals of $$\Gamma $$ Γ , i.e. they fulfill the conditions for selecting real regular finite-gap KP-II solutions in Dubrovin and Natanzon (Izv Akad Nauk SSSR Ser Mat 52:267–286, 1988). Since the soliton data are described by points in $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}$$ S M TNN , we establish a bridge between real regular finite-gap KP-II solutions (Dubrovin and Natanzon, 1988) and real regular multi-line KP-II solitons which are known to be parameterized by points in $$Gr^{\text{ TNN }}(k,n)$$ G r TNN ( k , n ) (Chakravarty and Kodama in Stud Appl Math 123:83–151, 2009; Kodama and Williams in Invent Math 198:637–699, 2014). We use the geometric characterization of spaces of relations on plabic networks introduced in Abenda and Grinevich (Adv Math 406:108523, 2022; Int Math Res Not 2022:rnac162, 2022. https://doi.org/10.1093/imrn/rnac162) to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in Lam (in: Current developments in mathematics, International Press, Somerville, 2014) for the computation of scattering amplitudes for on-shell diagrams $$N=4$$ N = 4 SYM (Arkani-Hamed et al. in Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, 2016) and govern the totally non-negative amalgamation of the little positive Grassmannians, $$Gr^{\text{ TP }}(1,3)$$ G r TP ( 1 , 3 ) and $$Gr^{\text{ TP }}(2,3)$$ G r TP ( 2 , 3 ) , into any given positroid cell $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}\subset {Gr^{\text{ TNN }} (k,n)}$$ S M TNN ⊂ G r TNN ( k , n ) . In our setting they control the reality and regularity properties of the KP-II divisor. Finally, we explain the transformation of both the curve and the divisor both under Postnikov’s moves and reductions and under amalgamation of positroid cells, and apply our construction to some examples.