AI 中文总结
本文从数学角度研究任意图（非平面）上的壳形式，推广平面情形的组合工具，证明一类壳形式的行列式公式，并揭示其与超树除子的联系。
AI 中文摘要
壳形式是格拉斯曼流形上的微分形式，起源于粒子物理学。它们通过具有 $n$ 个特殊“边界”顶点的二分图来定义。对壳形式的数学研究主要集中于图是平面的情形，此时可以利用Postnikov在研究全非负格拉斯曼流形时开创的组合工具。在本文中，我们研究任意图的壳形式。首先讨论如何将平面情形中的各种工具推广到任意图。然后，我们证明了一类首次出现在物理学文献中的壳形式的行列式公式。我们解释了这类形式与Castravet--Tevelev引入的 $M_{0,n}$ 的超树除子之间的关系。
英文摘要
On-shell forms are differential forms on the Grassmannian which arise in particle physics. They are defined using bipartite graphs with $n$ distinguished ``boundary'' vertices. Mathematical investigation of on-shell forms has largely focused on the case where the graph is planar, in which case one can utilize combinatorial tools pioneered by Postnikov in the study of the totally nonnegative Grassmannian. In this article, we investigate on-shell forms for arbitrary graphs. We first discuss how to extend various tools from the planar case to arbitrary graphs. We then prove a determinantal formula for a class of on-shell forms which first appeared in physics literature. We explain the relation between this class of forms and the hypertree divisors of $M_{0,n}$, introduced by Castravet--Tevelev.
Comments44 pages, 13 figures, comments welcome!
