Title: Wilson loops in SYM $N=4$ do not parametrize an orientable space Show Chinese title

Title: SYM $N=4$中的威尔逊环不参数化一个可定向空间

Abstract: 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. Show Chinese abstract

Abstract: 本文探讨了由 SYM $N=4$的树级威尔逊圈参数化的几何空间。我们证明，这个空间可以被视为定义在格拉斯曼流形的全非负子空间 $\mathcal{W}_{k,cn}$上的一个向量丛。此外，我们明确展示了在大多数情况下该丛是非定向的，并推测在剩余的情况下也是如此。利用格拉斯曼流形的 Deodhar 分解的组合学，我们识别出子空间 $\Sigma(W) \subset \mathcal{W}_{k,n}$，对于这些子空间，限制的丛位于正格拉斯曼流形之外。最后，在研究 Deodhar 分解的组合学时，我们给出了一个图解算法，用于读取确定每个 Deodhar 组件作为半代数集的方程。