标题： 一阶以上典范微分方程 显示英文标题

标题： Canonical Differential Equations Beyond Genus One

摘要： 我们首次讨论了双椭圆费曼积分的规范微分方程。 我们研究了双椭圆Lauricella函数，其中包括特别的是两圈非平面双盒的最大切割，已知这涉及一个亏格为二的双椭圆曲线。 我们具体考虑了三个和四个参数的Lauricella函数，每个都关联到一个亏格为二的双椭圆曲线，并构造了它们的规范微分方程。 虽然这一构造的核心步骤依赖于现有的方法 $\unicode{x2014}$ ，我们展示了这些方法在高亏格情况下也是适用的 $\unicode{x2014}$ ，我们使用关于规范形式下积分族的扭曲上同调相交矩阵结构的新思想，以更好地理解出现的新函数。 我们进一步观察到Siegel模形式出现在 $\varepsilon$-因子化微分方程矩阵中，很好地推广了从椭圆情况中的类似观察。 显示英文摘要

摘要： We discuss for the first time canonical differential equations for hyperelliptic Feynman integrals. We study hyperelliptic Lauricella functions that include in particular the maximal cut of the two-loop non-planar double box, which is known to involve a hyperlliptic curve of genus two. We consider specifically three- and four-parameter Lauricella functions, each associated to a hyperelliptic curve of genus two, and construct their canonical differential equations. Whilst core steps of this construction rely on existing methods $\unicode{x2014}$ that we show to be applicable in the higher-genus case $\unicode{x2014}$ we use new ideas on the structure of the twisted cohomology intersection matrix associated to the integral family in canonical form to obtain a better understanding of the appearing new functions. We further observe the appearance of Siegel modular forms in the $\varepsilon$-factorized differential equation matrix, nicely generalizing similar observations from the elliptic case.