Renormalisation under the prism of meromorphic functions in several variables

多变量亚纯函数棱镜下的重正化

• 批准号: 512473164
• 负责人: Professorin Dr. Sylvie Paycha, Ph.D.
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/512473164?language=en
• 关键词: Renormalisation; under; prism; meromorphic; functions

Meromorphic functions are central tools in the context of renormalisation and meromorphic functions in several variables with linear poles naturally arise from a multiparameter renormalisation. They are ubiquous in mathematics and physics, where they appear in various disguises such as Feynman integrals, discrete sums on cones and multizeta functions. Thus their study touches on various areas of mathematics, including number theory, toric geometry, category theory and infinite-dimensional topology. When compared with the more commonly used one parameter renormalisation, a multiparameter renormalisation operates a shift from the study of the (co-) algebraic structure underlying the subdivergences to be renormalised to the study of the pole structure of the resulting meromorphic germs. Hence, algebras of meromorphic functions in several variables provide a common tool to handle various renormalisation problems. In this framework, the renormalisation takes place in form of a generalised evaluator, a linear form on an algebra of meromorphic functions, which extends the ordinary evaluation at a point to an appropriate evaluation at poles. A classification of the generalised evaluators thanks to a group playing the role of a renormalisation group is the first main objective of this project. Such a classification can be carried out thanks to an underlying locality relation, a symmetric binary relation on meromorphic germs which has a separating role, serving the purpose of separating the subdivergences to be renormalised. Matching the needs of a multiparameter renormalisation requires enhancing the algebraic locality framework developed in previous work to a topological locality setup as well as generalising the related analytic tools beyond the orthogonality binary relation, which is the second challenging objective in this project. Our third aim is the study of the pole structure of regularised amplitudes derived from graphs decorated by holomorphic families of classical pseudodifferential operators and to capture their pole structure by universal properties of graphs using PROPs and their generalisations.

亚纯函数是重整化和多元亚纯函数中具有线性极点的重要工具 自然地由多参数重整化产生。它们在数学和物理学中无处不在，它们以各种伪装出现，如费曼积分，锥上的离散和和多zeta函数。因此，他们的研究涉及数学的各个领域，包括数论，环面几何，范畴论和无限维拓扑。当与更常用的一个参数重整化相比，多参数重整化操作的转移，从研究的（共）代数结构的subdivergences要重整的研究所产生的亚纯芽的极点结构。因此，多元亚纯函数代数提供了一个通用的工具来处理各种重整化问题。在这个框架中，重正化发生在一个广义的评估，一个线性形式的代数亚纯函数，它扩展了普通的评价在一个点上的适当的评价极点。该项目的第一个主要目标是通过一个扮演重新规范化小组角色的小组对一般评估人员进行分类。这样的分类可以进行感谢一个潜在的局部性关系，对称的二元关系的亚纯芽具有分离的作用，服务于分离的subdivergences要重新正规化。匹配的多参数重正化的需要，需要加强在以前的工作中开发的代数局部性框架的拓扑局部性设置，以及概括相关的分析工具超越正交二元关系，这是在这个项目中的第二个具有挑战性的目标。我们的第三个目标是研究正则振幅的极点结构，这些振幅来自于由经典伪微分算子的全纯族装饰的图，并利用PROPs及其推广来捕获它们的极点结构。