Algebraic Topology & Quantum Field Theory

在这个项目中，我们寻找一个代数定义，这将有助于数学家在量子场论涉及的几个分支工作。 这个想法是联合收割机的想法，从代数和拓扑学，服务于描述一个流形与庞加莱对偶。应用Hochshild结构导致代数模型的自由循环空间的流形和丰富的供应操作从弦拓扑。BV形式主义的出现和解决方案的量子主方程有望出现在几个上下文中，如微分，辛和全纯拓扑。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在理论物理学的语言中统一起来，即量子场论，但没有精确的数学概念为工作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会影响量子场论的真实的物理实例，相反，我们使用量子场论来帮助数学家，只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.