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

### Susama Agarwala

United States Naval Academy, Annapolis, USA### Cameron Marcott

Uniersity of Waterloo, Canada

## 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.

## Cite this article

Susama Agarwala, Cameron Marcott, Wilson loops in SYM $\mathcal{N}=4$ do not parametrize an orientable space. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), no. 4, pp. 583–622

DOI 10.4171/AIHPD/111