AI 中文总结
本文通过闵可夫斯基问题将朗道奇点与凸多胞形及正则细分相联系，揭示其与PL函数凸性及平面网的对偶关系。
AI 中文摘要
我们重新审视L. D. Landau于1959年关于费曼图相关积分可能奇点的经典论文。聚焦于实设置，我们通过闵可夫斯基问题将一组$p$个入射动量表示为凸多胞形$Q$的加权法向量，从而将其与凸几何联系起来。然后，$Q$的一个正则多面体细分$\mathcal{P}$将$p$展示为由$\mathcal{P}$的对偶图标记的朗道奇点，其中质量对应各面的面积。朗道/费曼/施温格乘子的正性被解释为PL函数的严格凸性。这给出了一类有趣的“多面体”朗道奇点。在可回溯至1959年原始论文的平面情形中，朗道图也可与Gaiotto-Moore-Witten的平面网相对应，后者提供了与正则多边形细分对偶的语言。
英文摘要
We revisit the classic 1959 paper of L. D. Landau on possible singularities of the integral associated to a Feynman graph. Focusing on the real setup, we relate it to convex geometry by representing a collection $p$ of incoming momenta as the weighted normals of a convex polytope $Q$ via the Minkowski problem. Then, a regular polyhedral subdivision $\mathcal{P}$ of $Q$ exhibits $p$ as a Landau singularity labelled by the dual graph of $\mathcal{P}$ with masses being the areas of the faces. Positivity of the Landau/Feynman/Schwinger multipliers is interpreted as strict convexity of a PL-function. This gives an interesting class of ``polyhedral'' Landau singularities. In the planar case going back to the original 1959 paper, Landau graphs can also be identified with plane webs of Gaiotto-Moore-Witten that provide a language dual to that of regular polygonal subdivisions.
Comments27 pages, 9 figures
