Wilson loops in SYM $\mathcal{N}=4$ do not parametrize an orientable space

Abstract

We explore the geometric space parametrized by (tree level) Wilson loops in SYM $\mathcal{N}=4$. We show that this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, ${\mathcal{W}}_{k,n}$. 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.