A study in $\mathbb{G}_{\mathbb{R}, \geq 0}$: from the geometric case book of Wilson loop diagrams and SYM $N=4$

Abstract

We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in the gauge theory of Supersymmetric Yang Mills (SYM) N = 4. By applying the tools developed to study total positivity in the real Grassmannian, we are able to systematically compute with all Wilson loop diagrams of a given size and find unexpected patterns and relationships between them. We focus on the smallest nontrivial multi-propagator case, consisting of 2 propagators on 6 vertices, and compute the positroid cells associated to each diagram and the homology of the subcomplex they generate in G R,≥0 (2,6). We also verify in this case the conjecture that the spurious singularities of the volume functional do all cancel on the codimension 1 boundaries of these cells.

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.…”

“…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. Specifically here we wish to examine whether it is possible to use WLDs to define a tessellation of the amplituhedron or more generally a tessellation of any "good" geometrical shape, whereby "good" means it only has a physical boundary (corresponding to poles of the amplitude) without any spurious boundaries.…”

An Algorithm to Construct the Le Diagram Associated to a Grassmann Necklace

“…(3) After performing step (2) The squares which should contain a + label are specified by Algorithm 2: Placing + labels in these squares and 0 labels in all remaining squares, we obtain the Le diagram The reader is invited to verify that applying Algorithm 1 to L(I) above yields exactly the Grassmann necklace in (A).…”

“…Indeed, the two main current techniques used in calculating scattering amplitudes in SYM N = 4 are BCFW recursion and Wilson loop diagrams, both of which make use of the combinatorial machinery of positroids [8]. In particular, one finds that the Le diagrams are crucial for understanding the geometry underlying the Wilson loop diagrams [2]. In a forthcoming paper, we study the combinatorics of the positroids associated to Wilson loop diagrams, and show that the natural objects to use in this setting are the Grassmann necklaces.…”

Quotients of uniform positroids

“…Introduced in [Pos06], positroids have proven to be a combinatorially exciting family of matroids. Positroids have rich connections to total positivity [KW11], cluster algebras [PS14], and physics [AF18]. There are several ways to describe positroids combinatorially [Oh11] either via Grassmann necklaces, decorated permutations, Le-diagrams, among other combinatorial objects as stated in [Pos06].…”