Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes

Abstract

Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM = 4 theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) corresponds to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.

Combinatorics of the geometry of Wilson loop diagrams II: Grassmann necklaces, dimensions, and denominators

“…For instance, square moves and merge-expand moves on the plabic graph give different graphical representations, hence parametrizations, of the same positroid cell in the Amplituhedron case. On the other hand, as we show in our companion paper [6], retriangulations of polygon dissections associated to Wilson loop diagrams give all the Wilson loop diagrams associated to the same positroid cell. Each distinct Wilson loop diagram associated to the same positroid cell gives a different parametrization [27].…”

“…19 Remark 3. 6 Given = (P, [ ]) and ∈ [ ], the order in which the propagators contribute to the algorithm during the construction of defines an order on P.…”

“…That is, the space defined by C(W) and the cell Σ(W) agree up to a set of measure 0. Most of the work in this paper and in the previous paper in the series [6] focuses on characterizing the positroid cell associated to each Wilson loop diagram W.…”

“…Remark 2.5 While this paper primarily approaches proofs from a combinatorial point of view, readers with a background in matroid theory are encouraged to keep the matroidal viewpoint in mind throughout; we expect that the two approaches will be complementary and will build upon each other. See the first paper in this series [6,Section 3] for some related results concerning the positroids of Wilson loop diagrams obtained by viewing them primarily as matroids.…”

“…Consequently, the spurious singularities are artifacts of how the integrals are represented and must cancel in the calculation of a tree-level scattering amplitude, although the details of how the cancellations come about remain unclear. The cancellation has been conjectured in general [24] and verified explicitly in a few cases [5,23], but it is hard to prove in general due to the fact that the relationship between Wilson loop diagrams and 3k-dimensional positroid cells of G R,≥0 (k, n) is neither one-to-one nor onto [6].…”