Two Wilson loop diagrams define the same positroid exactly when they differ by retriangulating certain exact subdiagrams, and inequivalent diagrams correspond to non-parallel faces of an associahedron.
Combinatorics of the geometry of Wilson loop diagrams II: Grassmann necklaces, dimensions, and denominators
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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. In this paper we study the structure of the associated positroids, as well as the structure of the denominator of the integrand defined by each diagram. We give an algorithm to derive the Grassmann necklace of the associated positroid directly from the Wilson loop diagram, and a recursive proof that the dimension of these cells is thrice the number of propagators in the diagram. We also show that the ideal generated by the denominator in the integrand is the radical of the ideal generated by the product of Grassmann necklace minors.
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Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes
Two Wilson loop diagrams define the same positroid exactly when they differ by retriangulating certain exact subdiagrams, and inequivalent diagrams correspond to non-parallel faces of an associahedron.