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

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.Comment: 47 pages, many figures; V2 and V3: minor modification

The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

The hypersimplex $\Delta_{k+1,n}$ is the image of the positive Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$ in $\mathbb{R}^n$. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}(Z)$ is the projection of the positive Grassmannian $Gr^{\geq 0}_{k,n}$ into $Gr_{k,k+2}$ under a map $\tilde{Z}$ induced by a matrix $Z\in \text{Mat}_{n,k+2}^{>0}$. Introduced in the context of scattering amplitudes, it is not a polytope, and has dimension $2k$. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first noted by Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -- images of positroid cells of $Gr^{\geq 0}_{k+1,n}$ under the moment map -- translate into sign conditions characterizing the T-dual Grasstopes -- images of positroid cells of $Gr^{\geq 0}_{k,n}$ under $\tilde{Z}$. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We prove the main conjecture of (LPW): a collection of positroid polytopes is a triangulation of $\Delta_{k+1, n}$ if and only if the collection of T-dual Grasstopes is a triangulation of $\mathcal{A}_{n,k,2}(Z)$ for all $Z$. Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip characterization of $\mathcal{A}_{n,k,2}(Z)$, and Lukowski--Parisi--Spradlin--Volovich's conjectures on $m=2$ cluster adjacency and on generalized triangles (images of $2k$-dimensional positroid cells which map injectively into $\mathcal{A}_{n,k,2}(Z)$). Finally, we introduce new cluster structures in the amplituhedron.Comment: 72 pages, many figures, comments welcome. v4: Minor edits v3: Strengthened results on triangulations and realizability of amplituhedron sign chambers. v2: Results added to Section 11.4, minor edit