• Corpus ID: 119295273

Wilson loops in SYM $N=4$ do not parametrize an orientable space

@article{Agarwala2018WilsonLI,
  title={Wilson loops in SYM \$N=4\$ do not parametrize an orientable space},
  author={Susama Agarwala and Cameron Marcott},
  journal={arXiv: Mathematical Physics},
  year={2018}
}

• S. Agarwala, Cameron Marcott
• Published 14 July 2018
• Mathematics
• arXiv: Mathematical Physics

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.