两个散射振幅的组合故事：参见两个双射
A Combinatorial Tale of Two Scattering Amplitudes: See Two Bijections
摘要：
在这篇论文中，我们将通过两个不同但并不完全相同的故事来探讨量子场论中散射振幅的组合学。第一部分讲述了$c_2$ -不变量的故事，这是一个与$ϕ^4$ -理论中的费曼积分相关的算术不变量，它研究了Kirchoff多项式和相关图多项式的零点。通过重新制定的$c_2$不变作为一个纯粹的组合问题，我们展示了如何枚举某些边二分区通过不动点自由对合可以完成一个特殊的情况下，长期追求后$c_2$完成猜想。第二部分讲述了正格拉斯曼和其胞元上的组合T-对偶映射的故事，这些映射与平面$\mathcal{N} = 4$ SYM理论中的散射振幅有关。特别是，T-对偶是超单形三角剖分和幅面三角剖分之间的桥梁，这两个对象表现为正格拉斯曼的图像。我们给出了一个算法，将T-对偶视为Le图上的一个映射，并描述了Le图的一个很好的结构（然后可以代替该算法）。通过这个Le图透视T-对偶，我们展示了如何直接解释地图两侧的正向细胞之间的维度关系。
英文摘要：
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.
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