多边形函数的GKZ系统的图解构造
Diagrammatic construction of GKZ systems for polygonal functions
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中文总结 AI 辅助
本文为多边形超几何函数构造GKZ超几何系统，利用图解算法和转移矩阵确定有限环面方程子系统，并给出实例验证。
中文摘要 AI 辅助
我们研究了最近引入的多边形超几何函数[arXiv:2502.12127, arXiv:2507.01904]，这些函数被猜想用于计算任意维度下的多点参数单圈共形积分。这些函数可以通过一个基于简单平面形状的图解算法系统地构造。给定一个$n$点多边形幂级数，我们构造了相应的GKZ超几何系统，该系统具有明确的图解解释。特别地，转移矩阵编码了所有相关的图解数据，它决定了相应的环面矩阵以及关系格的一组基，从而挑选出一个由$n(n-3)/2$个环面方程组成的有限子系统。这个有限子系统由二阶和三阶方程组成，而完整的环面族还包括高阶方程。我们通过几个重要的多边形函数实例来说明一般过程，包括第四类Appell函数和Srivastava--Daoust函数。
英文摘要
We investigate the recently introduced polygonal hypergeometric functions [arXiv:2502.12127, arXiv:2507.01904], which are conjectured to evaluate multipoint parametric one-loop conformal integrals in arbitrary dimensions. These functions can be systematically constructed using a diagrammatic algorithm that operates in terms of simple planar shapes. Given an $n$-point polygonal power series, we formulate the corresponding GKZ hypergeometric system, which admits an explicit diagrammatic interpretation. In particular, the transianic matrix, which encodes all the relevant diagrammatic data, determines the corresponding toric matrix along with a basis of the lattice of relations, thereby singling out a finite subsystem of $n(n-3)/2$ toric equations. This finite subsystem consists of second- and third-order equations, whereas the full toric family also includes higher-order ones. We illustrate the general procedure with several prominent examples of polygonal functions, including the fourth Appell and the Srivastava--Daoust functions.
发表机构
- I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute（P.N. 列别杰夫物理研究所 I.E. 塔姆理论物理系）
- Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科罗蒙诺索夫国立大学理论与数学物理研究所）
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