Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians

Abstract

The amalgamation of cluster varieties introduced by Fock and Goncharov in [21] plays a relevant role both in mathematical and physical problems. In particular, amalgamation in the totally non-negative part of positroid varieties is explicitly described as gluing of several copies of small positive Grassmannians, GrTP(1,3) and GrTP(2,3), has important topological implications [53] and naturally appears in the computation of amplitude scatterings in N=4 SYM theory [9,10]. Lam [44] has proposed to represent amalgamation in positroid varieties by equivalence classes of relations on bipartite graphs and to identify total non-negativity via appropriate edge signatures. In this paper we provide an explicit geometric characterization of such signatures in the setting of the planar bicolored trivalent directed perfect networks in the disk introduced in [52] to parametrize positroid cells SMTNN⊂GrTNN(k,n) using systems of relations for n-vectors.More precisely, to any such graph G, we associate a geometric signature satisfying both the full rank condition and the total non–negativity property on the full positroid cell. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation O and gauge ray direction l on G.The principal result is the enrichment of Postnikov's construction in [52] by associating measurements not only to boundary edges or vertices, but to internal edges as well. Indeed we generalize prior results by Postnikov [52] and Talaska [61] providing an explicit representation of the solution to the system of geometric relations on the network (N,O,l) of graph G and positive weights. At this aim, we assign canonical basis vectors in Rn at the boundary sinks and define the vectors components at the edge e as (finite or infinite) summations over the directed paths from e to the given boundary sink. Such edge vectors have the following properties:1) They solve the geometric system of relations on (N,O,l);2) Their components are rational in the weights with subtraction–free denominators, and have explicit expressions in terms of the conservative and edge flows of [61]. At the boundary sources they coincide with the entries of the boundary measurement matrix defined in [52]. If N is acyclically orientable, all components are subtraction–free rational expressions in the weights with respect to a convenient basis. Null edge vectors may occur on reducible networks not acyclically orientable;3) We provide explicit formulas both for the transformation rules of the edge vectors with respect to the orientation and the several gauges of the given network, and for their transformations due to moves and reductions of networks.Finally, we associate a Kasteleyn orientation to the graph following [17]. If the graph is bipartite, it is known that the partition functions of dimer configurations on the graph with given boundary conditions coincide with the Plucker coordinates of the corresponding point of the totally non-negative Grassmannian [54,44,60,1,8]. In the case of plabic graphs which are not bipartite we show that the partition function for a given boundary condition is not a multiple of the corresponding minor of the boundary measurement matrix. Therefore, in this case a statistical mechanical interpretation of the boundary measurement map remains open.