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.
Wilson loops in SYM $N=4$ do not parametrize an orientable space
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In this paper we explore the geometric space parametrized by (tree level) Wilson loops in SYM $N=4$. We show that, this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, $\mathcal{W}_{k,cn}$. Furthermore, we explicitly show that this bundle is non-orientable in the majority of the cases, and conjecture that it is non-orientable in the remaining situation. Using the combinatorics of the Deodhar decomposition of the Grassmannian, we identify subspaces $\Sigma(W) \subset \mathcal{W}_{k,n}$ for which the restricted bundle lies outside the positive Grassmannian. Finally, while probing the combinatorics of the Deodhar decomposition, we give a diagrammatic algorithm for reading equations determining each Deodhar component as a semialgebraic set.
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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.