We explore the geometric space parametrized by (tree level) Wilson loops in SYM . We show that this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, . 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 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.
Cite this article
Susama Agarwala, Cameron Marcott, Wilson loops in SYM do not parametrize an orientable space. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), no. 4, pp. 583–622DOI 10.4171/AIHPD/111