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Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk

Abstract

Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non–negative part of real Grassmannians (Postnikov et al. J. Algebr. Combin. 30(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) 92(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Abenda and Grinevich (2019). We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Abenda and Grinevich (Sel. Math. New Ser. 25(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.

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Acknowledgements

I thank Pavlo Pylyavskyy for stimulating my interest in Kasteleyn theorem, and Petr Grinevich for several useful discussions. I also thank the referee of the paper for valuable remarks.

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Open access funding provided by Alma Mater Studiorum - Università di Bologna within the CRUI-CARE Agreement.

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Correspondence to Simonetta Abenda.

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This research has been partially supported by GNFM-INDAM and RFO University of Bologna.

Communicated by: Michael Gekhtman

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Abenda, S. Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk. Math Phys Anal Geom 24, 35 (2021). https://doi.org/10.1007/s11040-021-09405-2

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Keywords

  • Totally non-negative Grassmannians
  • Positroid cells
  • Planar bipartite networks in the disk
  • Duality
  • Almost perfect matching
  • Kasteleyn signatures
  • M–curves
  • KP hierarchy
  • Real soliton and finite-gap solutions

Mathematics Subject Classification (2010)

  • 05C90
  • 14H70
  • 14M15
  • 37K40