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Real regular KP divisors on M-curves and totally non-negative Grassmanniansстатья
- Авторы: Abenda Simonetta, Grinevich Petr G.
- Журнал: Letters in Mathematical Physics
- Том: 112
- Номер: 6
- Год издания: 2022
- Издательство: Springer Nature
- Местоположение издательства: Switzerland
- DOI: 10.1007/s11005-022-01609-z
- Аннотация: In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell in the totally non-negative Grassmannian 𝐺𝑟^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 𝙼-curve, Γ, dual to the graph. The divisors are real regular KP-II divisors in the ovals of 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 𝐺𝑟^TNN (𝑘,𝑛), 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 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 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^TP (1,3) and 𝐺𝑟^TP (2,3), into any given positroid cell in 𝐺𝑟^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.
- Добавил в систему: Гриневич Петр Георгиевич