Feynman Motives

量子场论是高能物理和粒子物理中最复杂的预测计算技术。使用费曼图作为计算设备，可以获得涉及基本粒子和量子场的物理过程的高精度计算。尽管量子场论在粒子物理领域的成功应用有着悠久的历史，但它的数学仍然是神秘的，充满了美丽的挑战和开放的问题。数值证据表明，从发散的费曼积分中提取有限值的过程产生了一类数字，即多重zeta值，这对数论和代数几何具有重要意义。这表明量子场论与当前纯数学中一个重要的研究课题——格罗滕狄克的代数变量动机理论——之间存在着一种神秘的关系。这项研究计划的目的是了解这种关系的本质，并研究从中可以得出什么结果，既可以更好地理解使用代数几何工具进行费曼积分的非常困难的多环计算，也可以反过来理解我们如何使用量子场论扩展我们目前对动机的了解。正在研究的一个主要问题是，在可能减去散度之后，标量量子场论的费曼积分的计算何时会导致混合塔特动机的周期。利用费曼参数形式，这个问题反映了由费曼积分数据构造的仿射超曲面的相对上同调的动力性质。散度的减法编码在Hopf代数结构中，该结构本身与动机理论中自然出现的Hopf代数和对偶群有关。进一步理解量子场论与动机之间的关系所需要的主要步骤之一是将通过费曼图的超曲面的代数几何的更具体的方法与通过Tannakian范畴和Hopf代数的更抽象的方法相结合。

Quantum field theory is the most sophisticated technique for predictive computations in high-energy and particle physics. The use of Feynman diagrams as computational devices makes it possible to obtain high precision computations of physical processes involving elementary particles and quantum fields. Despite its long history of successful applications to the world of particle physics, the mathematics of quantum field theory is still mysterious and full of beautiful challenges and open problems. Numerical evidence suggests that the procedure of extracting finite values from divergent Feynman integrals gives rise to a class of numbers, multiple zeta values, that are of great significance to number theory and algebraic geometry. This suggests a mysterious relation between quantum field theory and an important research topic of current interest in pure mathematics: Grothendieck's theory of motives of algebraic varieties. The purpose of this research proposal is to understand the nature of this relation and investigate what results can be derived from it, both in terms of gaining some better understanding of the very difficult multi-loop computations of Feynman integrals using tools from algebraic geometry, and conversely of understanding how we can extend our current knowledge of motives using quantum field theory.One of the main questions under investigation is when, possibly after a subtraction of divergences, the computation of a Feynman integral for a scalar quantum field theory results in a period of a mixed Tate motive. Using the Feynman parametric form, this question reflects the motivic nature of a relative cohomology of an affine hypersurface constructed out of the data of the Feynman integral. The subtraction of divergences is encoded in a Hopf algebra structure, which is itself related to Hopf algebras and dual groups that appear naturally in the theory of motives.One of the main steps that are needed to further understand the relation between quantum field theory and motives is combining the more concrete approach via the algebraic geometry of hypersurfaces of Feynman graphs with the more abstract approach via Tannakian categories and Hopf algebras.