JournalsaihpdVol. 8, No. 4pp. 583–622

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

  • Susama Agarwala

    United States Naval Academy, Annapolis, USA
  • Cameron Marcott

    Uniersity of Waterloo, Canada
Wilson loops in SYM $\mathcal{N}=4$ do not parametrize an orientable space cover
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Abstract

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