Wilson Loop Diagrams and Positroids

Abstract

Abstract:In this paper, we study a new application of the positive Grassmannian to Wilson loop diagrams (or MHV diagrams) for scattering amplitudes in N= 4 Super Yang-Mill theory (N = 4 SYM). There has been much interest in studying this theory via the positive Grassmannians using BCFW recursion. This is the first attempt to study MHV diagrams for planar Wilson loop calculations (or planar amplitudes) in terms of positive Grassmannians. We codify Wilson loop diagrams completely in terms of matroids. This allows us to apply the combinatorial tools in matroid theory used to identify positroids (non-negative Grassmannians) to Wilson loop diagrams. In doing so, we find that certain non-planar Wilson loop diagrams define positive Grassmannians. While non-planar diagrams do not have physical meaning, this finding suggests that they may have value as an algebraic tool, and deserve further investigation.

The twistor Wilson loop and the amplituhedron

“…However the WLDs apparently lend themselves very naturally and directly to a geometrical interpretation and in this paper we wish to look again at the relationship between WLDs and the amplituhedron. Previous work also examining this connection includes [9,18,26]. In particular in [26] it was shown that the WLDs give a very natural description of the physical boundary of the amplituhedron.…”

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

“…Every element of (∅) defines a circuit of size 1, and all other circuits of ( ) must contain at least two elements. In this section we develop this analysis further, showing the converse of this result from [5] (see Theorem 3.19). We also enumerate the number of diagrams that map to a particular matroid in Corollary 3.20.…”

“…There are a few things to note here. First, what we call weakly admissible here is called admissible in ( [5], see definition 1.11 and section 1.3). Note also that if two propagators have the same pair of supporting edges, or if a propagator is supported on two adjacent edges, then the Wilson loop diagram is not admissible or even weakly admissible.…”

“…Comparing this to Theorem 2.6, we see that ( ) is also the matroid realized by ( ). As mentioned above, Theorem [3.38] of [5] shows that if is weakly admissible, the matroid ( ) is a positroid. Furthermore, Theorem 5.1 [18] shows that the space parametrized by the matrices ( ) matches the positroid cells defined by ( ), up to a set of measure 0.…”

“…The second paper [3] in this series identifies precisely which positroid cell is associated to a given In recent years, there has been an active program researching the geometry and combinatorics underlying SYM = 4 theory [5,8,13,14,16]. This body of work started with the observation that the on shell (tree level) amplitudes of this theory correspond to the volume of a subspace of a positive Grassmannian, called an Amplituhedron [8].…”