In perturbative quantum field theory one encounters certain, very specific geometries over the integers. These perturbative quantum geometries determine the number contents of the amplitude considered. In the article ‘Modular forms in quantum field theory’ F. Brown and the author report on a first list of perturbative quantum geometries using the $c_2$-invariant in $\varphi^4$ theory. A main tool was denominator reduction which allowed the authors to examine graphs up to loop order (first Betti number) 10. We introduce an improved quadratic denominator reduction which makes it possible to extend the previous results to loop order 11 (and partially orders 12 and 13). For comparison, also $\varphi^4$ graphs are investigated. Here, we extend the results from loop order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously 157)—while being consistent with all major $c_2$-conjectures—leads to a more refined picture of perturbative quantum geometries. In the appendix, Friedrich Knop proves a Chevalley–Warning–Ax theorem for double covers of affine space.

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微扰量子场论中的几何

在微扰量子场论中，人们会遇到某些非常具体的整数几何。这些微扰量子几何决定了所考虑的振幅的数量内容。在“量子场论中的模形式”一文中，F. Brown 和作者报告了使用 $\varphi^4$ 理论中的 $c_2$-不变量的微扰量子几何的第一个列表。一个主要的工具是分母减少，它允许作者检查循环顺序（第一个 Betti 数）10 的图。我们引入了改进的二次分母减少，这使得可以将之前的结果扩展到循环顺序 11（和部分顺序 12 和部分） 13)。为了比较，还研究了 $\varphi^4$ 图。在这里，我们将结果从循环顺序 9 扩展到 10。包含 4801 个独特的 $c_2$-不变量（以前为 157 个）的新数据库——同时与所有主要的 $c_2$-猜想一致——导致对微扰量子几何的更精确描述。在附录中，Friedrich Knop 证明了仿射空间双覆盖的 Chevalley-Warning-Ax 定理。