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 N = 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) correspond 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.

Cancellation of spurious poles in N=4 SYM: physical and geometric

“…In other words, the polynomial R(V) corresponds to the product of the frozen variables of the cluster algebra associated to Σ in the choice of coordinates given by M V . Recent work has also shown that the polynomial R(V(P)) is a special case of this phenomenon [9]. For I(V(P)), the Grassmann necklace of Σ(V(P)), we have R(V(P)) = rad( I i ∈I(V(P)) ∆ I i (x)).…”

“…Note that the polynomial R(V) is dependent on the minimal parameterization of the positroid defined by the Grassmann necklace I. For instance, in [9], the authors give conditions for when multiple Wilson loop diagrams can correspond to the same positroid cell. For example, the two by six matrices defined by V 1 = {{1, 2, 4, 5}, {1, 2, 3, 4}} and V 2 = {{1, 2, 4, 5}, {2, 3, 4, 5}} both correspond to the positroid variety with Grassmann necklace I = {12, 23, 34, 45, 51, 12}.…”

“…We indicate these polynomials as R(V(P)). Each Wilson loop diagram represents an N k M HV diagram, and also corresponds to a convex subspace of G R,≥0 (k, n) called a positroid cell, denoted Σ(V(P)) [8,9]. Any positroid cell, Σ, can also be defined by a set of Plücker coordinates called the Grassmann necklace, I[10, Section 16].…”

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

“…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 [5] focuses on characterizing the positroid cell associated to each Wilson loop diagram W .…”

“…SYM N = 4 is a finite theory [1,2,12], so the singularities must cancel out in the sum over all diagrams, though how this works out mathematically is unclear. The cancellation has been conjectured in general [20] and verified explicitly in a few cases [3,19], 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 [5].…”