Motivic Geometry of two-Loop Feynman Integrals

IF 0.6 4区 数学 Q3 MATHEMATICS
Charles F Doran, Andrew Harder, Pierre Vanhove, Eric Pichon-Pharabod
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引用次数: 0

Abstract

We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch (Double box motive. SIGMA 2021;17,048) that in the well-known double-box example, there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the non-planar two-loop tardigrade graph is that of a K3 surface. In an appendix by Eric Pichon-Pharabod, we argue via high-precision numerical computations that the Picard number of this K3 surface is generically 11 and we compute the expected lattice polarization. Lastly, we show that generic members of the ice cream cone family of graph hypersurfaces correspond to the pairs of sunset Calabi–Yau varieties.
双环费曼积分的动机几何学
我们研究了一般物理参数下二环费曼积分所附立方超曲面的几何和霍奇理论。我们表明,平面双环费曼图上的霍奇结构分解为混合塔特片段和超椭圆、椭圆或有理曲线族的霍奇结构,这取决于时空维度。对于有少量边的双环图,我们给出了更精确的结果。特别是,我们恢复了布洛赫的一个结果(双箱动机。 SIGMA 2021;17,048),即在著名的双箱例子中,存在一个潜在的椭圆曲线族,我们给出了这些椭圆曲线的具体描述。我们证明了非平面双环迟行图的动机是 K3 曲面。在埃里克-皮雄-帕拉博德(Eric Pichon-Pharabod)的附录中,我们通过高精度数值计算论证了这个 K3 曲面的皮卡数一般为 11,并计算了预期的晶格极化。最后,我们证明了冰激凌锥系列图超曲面的一般成员对应于日落卡拉比优变体对。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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