Vertex algebras and geometry of manifolds

这个项目的目的是加深对顶点代数和流形几何之间关系的理解。这种关系是最近发现的，是理论物理和数学之间经典关系的无维版。事实上，流形上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罗姆复形的研究。试图将其推广到流形上的环空间的情况下，已经创建了一个全新的数学和物理概念宇宙。一个这样的概念，顶点代数，弦理论和无限维表示理论的子理论，已经被证明在抽象有限群论等遥远的数学物理领域中是不可或缺的。最近，由于戈尔布诺夫和马利科夫等人的工作，顶点代数已卓有成效地应用于几何的几个章节，包括椭圆亏格和镜像对称。弦理论是量子场论的一部分，也是唯一存在的“万物理论”的候选者，尽管尚未达到其最终目标，但它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。顶点代数是一个同时推广了李代数和交换代数概念的概念，就是一个明显的例子。很早以前，它就在诸如表示论、群论、组合学和数论等古老而多样的领域中找到了应用。本项目致力于探索最近发现的与高维空间几何的一种关系。

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.