Real regular KP divisors on M-curves and totally non-negative Grassmannians

In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell STNN M in the totally non-negative Grassmannian GrTNN(k, n) to the spectral data for the relevant class of real regu lar 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 M-curve, , dual to the graph. The divisors are real regular KP-II divisors in the ovals of , 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 STNN M , 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 GrTNN(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 intro duced 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 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, GrTP(1, 3) and GrTP(2, 3), into any given positroid cell STNN M ⊂ GrTNN(k, n). In our set ting 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 con struction to some examples

Real Soliton Lattices of the Kadomtsev Petviashvili II Equation and Desingularization of Spectral Curves: The GrTP(2, 4) Case

We apply the general construction developed in our previous papers to the first nontrivial case of GrTP(2, 4). In particular, we construct finite-gap real quasi-periodic solutions of the KP-II equation in the form of a soliton lattice corresponding to a smooth M-curve of genus 4 which is a desingularization of a reducible rational M-curve for soliton data in GrTP(2, 4

Rational Degenerations of M-Curves, Totally Positive Grassmannians and KP2-Solitons

We establish a new connection between the theory of totally positive Grassmannians and the theory of M-curves using the finite-gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev\u2013Petviashvili 2 equation [see (1)], which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian GrTP(N,M) a reducible curve which is a rational degeneration of an M-curve of minimal genus g=N(M 12N), and we reconstruct the real algebraic-geometric data \ue1 la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction, it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth M-curves. In our approach, we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection GrTP(r+1,M 12N+r+1)\u21a6GrTP(r,M 12N+r