Feynman Motives

量子场论是高能物理和粒子物理学中最复杂的预测计算技术。费曼图作为计算设备的使用使得获得涉及基本粒子和量子场的物理过程的高精度计算成为可能。尽管量子场论在粒子物理学领域有着悠久的成功应用历史，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Feynman积分中提取有限值的过程产生了一类对数论和代数几何具有重要意义的数，即多重Zeta值。这暗示了量子场论与当前纯数学领域感兴趣的一个重要研究课题之间的神秘联系：格罗森迪克的代数变种动机理论。这项研究的目的是了解这种关系的本质，并研究它可以得出什么结果，这既是为了更好地理解使用代数几何中的工具进行费曼积分的非常困难的多循环计算，也是为了了解我们如何使用量子场论来扩展我们当前的动机知识。正在研究的主要问题之一是，在可能减去发散之后，标量量子场论的费曼积分的计算何时会产生混合状态动机。利用Feynman参数形式，这个问题反映了由Feynman积分数据构造的仿射超曲面的相对上同调的动机性质。散度的减法被编码在Hopf代数结构中，这种结构本身与Hopf代数和在动机理论中自然出现的对偶群有关。进一步理解量子场论和动机之间的关系所需要的主要步骤之一是将通过Feynman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.