A study in ${\mathbb{G}}_{\mathbb{R},\ge 0}(2,6)$: 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 Supersymmetric Yang Mills (SYM) $N=4$. In particular, we study the smallest non-trivial multi-propagator case, consisting of 2 propagators on 6 vertices. We do this by translating the integrals of the theory to the combinatorics of the positive geometry each diagram represents, specifically identifying the positroid cells defined by each diagram and the homology of the subcomplex they collectively generate in ${\mathbb{G}}_{\mathbb{R},\ge 0}(2,6)$. We verify the conjecture that the spurious singularities of the volume functional do all cancel on the codimension 1 boundaries of these cells, in this case. We also show that how the spurious singularities cancel is actually much more complicated than previously understood. The direct calculation laid out in this paper identifies many intricacies and artifacts of the geometry of Wilson loop diagram that need further study.