Algebraic structures in perturbative quantum field theory

摘要奖：DMS-1007168首席研究员：凯文·J·科斯特罗拟议的研究涉及到一种基于作用化代数概念的微扰量子场论的新方法。因式分解代数是丰富的代数对象，同时推广了E-n代数和手征代数的概念。这个提议的目的是证明微扰重整化技术允许人们从定义物理系统的拉格朗日开始构造因式分解代数。这种方法在量子场论中的几个应用也被提出，包括与椭圆上同调和Witten基因组有关的应用。量子场论是理论HySICS的基本工具，并支持物理学家目前对基本粒子行为的理解。此外，量子场论对数学的发展也产生了巨大的影响。这项研究计划的目的是为量子场论提供数学基础，并将这些基础应用于与物理有关的几何问题。

AbstractAward: DMS-1007168Principal Investigator: Kevin J. CostelloThe proposed research is concerned with a new approach to perturbative quantum field theory, based on the concept of a actorization algebra. Factorization algebras are rich algebraic objects which simultaneously generalize the notion of E-n algebra and that of chiral algebra. The aim of this proposal is to show that the techniques of perturbative renormalization allow one to construct factorization algebras starting from a Lagrangian defining a physical system. Several applications of this approach to quantum field theory are also being proposed, including one related to elliptic cohomology and the Witten genus.Quantum field theory is a fundamental tool in theoretical hysics, and underpins physicists current understanding of the behaviour of elementary particles. Quantum field theory has additionally had a great influence on the development of mathematics. The aim of this research proposal is to provide mathematical foundations for quantum field theory, and to apply these foundations to problems in geometry related to physics.