Algebraic Topology & Quantum Field Theory

在这个项目中，我们寻找一个代数定义，它将服务于在量子场论触及的几个分支中工作的数学家。这个想法是将代数和拓扑学的思想结合起来，用来描述具有庞加莱对偶性的流形。应用Hochshild结构得到了流形的自由环空间的代数模型和来自弦拓扑的丰富的运算供应。BV形式出现了，量子主方程的解有望出现在几种背景下，如微分、辛拓扑和全纯拓扑学。这个项目在更实际的层面上的动机是，几个非常有趣的数学讨论以理论物理的语言统一起来，即量子场论，但没有精确的数学概念为工作的数学家提供这样的综合。而且，一旦我们有了这样的基本数学概念，数学家就能够系统地发展它们的结构，并深入到更深层次。我预计这项工作不会影响量子场论的真实物理例子，而是反过来--我们使用量子场论来帮助数学家，只需试图定义它的某一部分。

In this project we search for an algebraic definition that will serve the mathematician working in several of those branches touched by quantum field theory. The idea is to combine ideas from algebra and topology that serve to describe a manifold with its Poincare' duality. Applying Hochshild constructions leads to algebraic models of the free loop space of the manifold and the rich supply of operations from string topology. The BV formalism appears and solutions of the quantum master equation hopefully appear in several contexts such as differential, symplectic and holomorphic topology. The motivation for this project at a more practical level is that several extremely interesting mathematical discussions are united in the language of theoretical physics, namely quantum field theory, but there are not precise mathematical concepts which provide such a synthesis for the working mathematician. Also once we have such underlying mathematical concepts, mathematicians will be able to develop their structure in a systematic manner and go deeper. I do not expect this work to impact the real physical examples of quantum field theory, but rather the converse-we use quantum field theory to help mathematicians by merely trying to define some part of it.