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工作的数学家。这个想法是结合代数和拓扑学的思想，用来描述具有庞加莱对偶性的流形。利用hochschild构造可以得到流形自由循环空间的代数模型，并提供了丰富的弦拓扑运算。在微分拓扑、辛拓扑和全纯拓扑等环境中，出现了BV形式，并有望出现量子主方程的解。在更实际的层面上，这个项目的动机是几个非常有趣的数学讨论在理论物理的语言中联合起来，即量子场论，但是没有精确的数学概念为工作的数学家提供这样的综合。此外，一旦我们有了这些基本的数学概念，数学家将能够以系统的方式发展它们的结构，并深入研究。我并不期望这项工作影响量子场论的实际物理例子，相反，我们使用量子场论来帮助数学家，仅仅是试图定义它的一部分。

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.