AI 中文总结
该论文研究正合流形的奇点判定方法，给出两种计算T-不动点重数的途径，关联其与平面N=4超对称杨-米尔斯理论振幅的相关性，是2024年8月论文的修订版。
AI 中文摘要
正合流形Π_f对格拉斯曼流形Gr(k,n)进行了分解；它们可通过有界仿射置换f枚举，这类置换与诸多有趣的组合对象存在双射关系。此外，正合流形的非负部分参数化了计算平面N=4超对称杨-米尔斯理论振幅时的积分空间。本论文解答的核心问题是：正合流形Π_f是否存在几何奇点。我们证明只需在T-不动点λ处检查奇点，且可通过计算限制到该点的Π_f的等变上同调得到这些点λ处的重数。我们给出两种实现方法：(1) 利用仿射管道图的图示方法；(2) 计算方法。对于方法(2)，我们已编写代码完成计算并输出重数（相关文件见此http URL）。我们还列出了n≤6时正合流形上所有点的重数表。此外，我们描述了由删除/收缩给出的对(Π_f,λ)的序关系，该序关系与光滑性相互作用良好，并阐述了仿射管道图与有限管道图之间的关系。论文第二部分与平面N=4 SYM的物理学相联系。我们简要介绍了量子场论、主奇点及N=4 SYM中的正合流形。随后解释了Britto-Cachazo-Feng-Witten (BCFW)递归与树图振幅的BCFW桥分解。我们描述了如何利用BCFW桥在管道图上构建树图振幅。最后，我们研究了与树图振幅的逆软因子相关的组合问题，并探讨正合流形中的奇点是否与振幅存在关联。本文是2024年8月撰写的原论文的修订版。
英文摘要
Positroid varieties $Π_f$ provide a decomposition of the Grassmannian $Gr(k,n)$; they can be enumerated using bounded affine permutations ($f$) which have bijections with a number of interesting combinatorial objects. Furthermore, (the nonnegative part of) positroid varieties parameterize the space that is integrated over when calculating amplitudes in planar N=4 supersymmetric Yang-Mills theory. The main question we answer in this thesis is whether a positroid variety $Π_f$ has any geometric singularities. We show that it is sufficient to check singularity at the $T$-fixed points ($λ$) and we can obtain the multiplicity at these points $λ$ by calculating the equivariant cohomology of $Π_f$ restricted to the point. We give 2 ways of doing this: (1) A diagrammatic way using affine pipe dreams and (2) A computational method. For (2), we have written code that does the computation and outputs the multiplicity (files at josephflueg.github.io). We have included tables listing the multiplicities of all the points on positroid varieties up to $n=6$. We also describe an ordering on pairs $(Π_f,λ)$ given by deletion/contraction that interacts nicely with smoothness, and describe the relationship between affine pipe dreams and finite pipe dreams. In Part II of this thesis connects with physics of planar N=4 SYM. We briefly introduce quantum field theory, leading singularities, and positroids in N=4 SYM. We then explain Britto-Cachazo-Feng-Witten (BCFW) recursion and the BCFW bridge decomposition of an on-shell diagram. We describe how to build an on-shell diagram on a pipe dream using BCFW bridges. Finally, we work out some combinatorics related to inverse soft factors for on-shell diagrams and explore whether singularities in positroid varieties have relevance to amplitudes. This is a revised version of my thesis originally written in August 2024.
CommentsPh.D. Dissertation, Cornell University, 2024 (Revised 2026)
