Vertex algebras and geometry of manifolds

顶点代数和流形几何

• 批准号: 0500573
• 负责人: Feodor Malikov
• 金额: --
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500573&HistoricalAwards=false
• 关键词: Vertex; algebras; geometry; manifolds

The purpose of this project is to further the understanding of therelation between vertex algebras and the geometry of manifolds.This relation, discovered relatively recently, is an infinitedimensional version of the classical relation between theoreticalphysics and mathematics. Indeed, quantum mechanics of a particleon a manifold is a part of analysis on manifolds. Similarly,super-symmetric quantum mechanics is essentially equivalent tothe study of the de Rham complex on manifolds. Attempts to carrythis over to the case of the space of loops on a manifold havecreated an entirely new Universe of mathematical and physicalconcepts. One such concept, vertex algebra, a child of stringtheory and infinite dimensional representation theory, has alreadyproved indispensable in areas as remote form mathematical physicsas abstract finite group theory. Relatively recently, thanks towork of Gorbounov and Malikov among others, vertex algebras havebeen fruitfully applied to several chapters of geometry includingelliptic genus and mirror symmetry.String theory, a part of quantum field theory and the onlyexisting candidate for the "theory of everything", while havingnot yet achieved its ultimate goal, has enriched modernmathematics and physics with a dazzling array of new concepts andnew relations among old concepts. Vertex algebra, a concept thatsimultaneously generalizes the notions of Lie and commutativealgebra, is one conspicuous example. It has long since foundapplications in areas as old and diverse as representation theory,group theory, combinatorics, and number theory. The presentproject is devoted to the exploration of a recently discoveredrelation to geometry of higher dimensional spaces.

这个项目的目的是进一步了解顶点代数和流形几何之间的关系。这种关系是最近才发现的，是理论物理和数学之间经典关系的无限维版本。实际上，流形粒子的量子力学是流形分析的一部分。同样，超对称量子力学本质上等同于流形上的德拉姆复形的研究。将这一理论应用于流形上的循环空间的尝试，创造了一个全新的数学和物理概念的宇宙。其中一个概念，顶点代数，弦理论和无限维表示理论的产物，已经被证明在远程数学物理和抽象有限群论等领域不可或缺。最近，多亏了Gorbounov和Malikov等人的工作，顶点代数在几何学的几个章节中得到了丰硕的应用，包括椭圆属和镜像对称。弦理论是量子场论的一部分，是“万有理论”的唯一候选者，虽然尚未达到其最终目标，但却以一系列令人眼花缭乱的新概念和旧概念之间的新关系丰富了现代数学和物理学。顶点代数是一个突出的例子，它同时推广了李代数和交换代数的概念。它早已在诸如表示理论、群论、组合学和数论等古老而多样的领域中得到了应用。本项目致力于探索最近发现的与高维空间几何的关系。