Application of methods of arithmetic geometry and homological algebra to quantum field theory and string theory

摘要奖：DMS-0805989首席研究员：阿尔伯特·施瓦茨这些项目中的第一个是关于p-ad方法在拓扑弦上的应用。这些项目的目标是用算术几何来表示物理量，并使用这些表达式来证明积分定理。另一组研究涉及量子场论的超对称变形。这是与M.Movshev的合作，应用了L-无穷大和A-无穷大代数的理论。另一个项目致力于分析弦理论和量子场论之间的关系。主要研究人员认为，量子场论可以公式化，使时间和空间不出现为主要概念，并计划证明弦理论在这个意义上可以被视为量子场论。这些研究项目基于现代数学方法在物理中的应用，特别是利用算术几何和同调代数来提高我们对量子场论和弦理论的理解。最初为其他目的而发展的理论性数学，如研究数论中的丢番图方程，似乎直接与理论物理中的挑战性问题有关。

AbstractAward: DMS-0805989Principal Investigator: Albert SchwarzThe first of these projects concerns applications of p-adicmethods to topological strings. The goal of these projects is toexpress physical quantities in terms of arithmetic geometry overp-adic numbers and to use these expressions to prove integralitytheorems. Another set of investigations deals withsupersymmetric deformations of quantum field theories. This isjoint work with M. Movshev and applies the theory of L-infinityand A-infinity algebras. Another project is devoted to theanalysis of the relation between string theory and quantum fieldtheory. The principal investigator has argued that quantum fieldtheory can be formulated such that time and space do not appearas primary notions, and is planning to show that string theorycan be regarded as a quantum field theory in this sense.These research projects are based on the application of methodsof modern mathematics to physics, particularly making use ofarithmetic geometry and homological algebra to improve ourunderstanding of quantum field theory and string theory.Sophisticated mathematics originally developed for other purposessuch as the study of Diophantine equations in number theory seemsto bear directly on challenging issues in theoretical physics.