论文标题
两个散射幅度的组合故事：请参阅两只徒
A Combinatorial Tale of Two Scattering Amplitudes: See Two Bijections
论文作者
论文摘要
在这篇论文中，我们经历了两个不同但并非不同的故事，其基本主题是量子场理论中散射幅度的组合。 第一部分讲述了$ c_2 $ invariant的故事，这是与$ ϕ^4 $ - 理论中的Feynman积分相关的算术不变的，它研究了Kirchoff多项式和相关图形的零。通过将$ C_2 $ invariant重新定义为一个纯粹的组合问题，我们展示了如何通过无定点的无定点参与来枚举某些边缘两部分，可以完成$ C_2 $完成猜想之后长期寻求的特殊情况。 第二部分讲述了其细胞上阳性草个子的故事和组合t偶（t偶）图，这与平面$ \ Mathcal {n} = 4 $ sym理论的散射幅度有关。尤其是，T偶性是axplituhedron的高刺和三角剖分的三角形之间的桥梁，这是两个作为阳性格拉斯曼尼亚阳性图像的物体。我们给出了一种算法，用于将T-偶数视为LE图上的地图，并将其表征为LE图的良好结构（然后可以用来代替该算法）。通过对T偶性的LE图观点，我们展示了如何直接解释地图两侧的阳性细胞之间的尺寸关系。
In this thesis, we take a journey through two different but not dissimilar stories with an underlying theme of combinatorics emerging from scattering amplitudes in quantum field theories. The first part tells the tale of the $c_2$-invariant, an arithmetic invariant related to the Feynman integral in $ϕ^4$-theory, which studies the zeros of the Kirchoff polynomial and related graph polynomials. Through reformulating the $c_2$-invariant as a purely combinatorial problem, we show how enumerating certain edge bipartitions through fixed-point free involutions can complete a special case of the long sought after $c_2$ completion conjecture. The second part tells the tale of the positive Grassmannian and a combinatorial T-duality map on its cells, as related to scattering amplitudes in planar $\mathcal{N} = 4$ SYM theory. In particular, T-duality is a bridge between triangulations of the hypersimplex and triangulations of the amplituhedron, two objects that appear as images of the positive Grassmannian. We give an algorithm for viewing T-duality as a map on Le diagrams and characterize a nice structure to the Le diagrams (which can then be used in lieu of the algorithm). Through this Le diagram perspective on T-duality, we show how the dimensional relationship between the positroid cells on either side of the map can be directly explained.
