Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk

TL;DR

This paper provides a geometric interpretation of Kasteleyn matrices on planar bipartite graphs, linking them to Grassmannians, positroid cells, and KP soliton solutions, offering new proofs and algebraic geometric insights.

Contribution

It introduces a geometric characterization of Kasteleyn signatures, connecting them to Grassmannian parametrizations and algebraic geometric data for KP solitons.

Findings

Kasteleyn signatures are geometric if and only if they satisfy certain relations.

The parametrization of positroid cells via Kasteleyn matrices matches Postnikov's boundary measurement map.

The algebraic geometric data associated with KP solitons is invariant and coincides with previous constructions.

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 (see Refs. [54,43,44,58,7]). 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 Ref. [5]. 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 Refs. [4,6] for the present class of graphs.