Homological Methods in Quantum Field Theory

量子场论中的同调方法

• 批准号: 0401433
• 负责人: Dmitry Tamarkin
• 金额: --
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401433&HistoricalAwards=false
• 关键词: Homological; Methods; Quantum; Field; Theory

The PI is planning to focus on several problems which are related with the procedure of quantization in quantum field theory and quantum mechanics.These problems include:1) quantization of coisson algebras;2) mathematically precise formulation of Batalin-Vilkovitski formalism;3) Action of Grothendieck-Teichmueller group on various formality and quantization morphisms.The first two problems arise in the quantum field theory. The problem of quantization of coisson algebras is posed by A. Beilinson and V. Drinfeld; they found a rather non-trivial solution to this problem in a particular case of linear coisson brackets. The general quantization problem is much harder and is not accessible by similar methods. An appropriate tool may be the deformation theory and introduction of an additional structure on an appropriate deformation complex.The Batalin-Vilkovitsky formalism is one of the most powerful tools in quantization of systems with sophisticated gauge symmetries. This formalism implies an extensive use of path integrals, whence the lack of mathematical meaning of the most important ingredients of the construction, such as the operator $\Delta$ and the $BV$-bracket. The difficulty in defining path integrals are well known, a straightforward extension of usual (finitely-dimensional) integration rules necessarily leads to divergencies. It is only a careful analysis of Batalin-Vilkovitski formalism involving a theory of D-modules and homological algebra that can allow one to construct a mathematically meaningful theory.The action of Grothendieck-Teichmueller group is known to be present on the set of quantization functors of Lie bialgebras as well as on the set of formality quasi-isomorphisms from Kontsevich's formality theorem. This fascinated subject was originated by V. Drinfeld and was further developed by P.Etingof-D. Kazhdan, M. Kontsevich and other authors. Yet there are several open questions concerning this action on the space of formality quasi-isomorphisms, namely, whether it is transitive or free in a certain homotopical sense. I hope that studying these problems should deepen our understanding of algebro-geometric and motivic aspects of quantization.

PI 计划重点研究与量子场论和量子力学中的量子化过程相关的几个问题。这些问题包括：1）柯松代数的量子化；2）Batalin-Vilkovitski 形式主义的数学精确表述；3）Grothendieck-Teichmueller 群对各种形式和量子化态射的作用。前两个问题出现在量子场论中。柯松代数的量化问题由 A. Beilinson 和 V. Drinfeld 提出；在线性 Coisson 括号的特殊情况下，他们找到了一个相当重要的解决方案。一般的量化问题要困难得多，并且无法通过类似的方法来解决。适当的工具可能是变形理论和在适当的变形复合体上引入附加结构。Batalin-Vilkovitsky 形式主义是具有复杂规范对称性的系统量化中最强大的工具之一。这种形式主义意味着路径积分的广泛使用，因此缺乏构造中最重要成分的数学意义，例如运算符 $\Delta$ 和 $BV$ 括号。定义路径积分的困难是众所周知的，通常（有限维）积分规则的直接扩展必然会导致发散。只有对涉及 D 模理论和同调代数的 Batalin-Vilkovitski 形式主义进行仔细分析，才能构建出数学上有意义的理论。已知 Grothendieck-Teichmueller 群的作用存在于李双代数的量化函子集合以及来自 Kontsevich 形式的形式准同构集合上 定理。这个令人着迷的主题由 V. Drinfeld 发起，并由 P.Etingof-D 进一步发展。 Kazhdan、M. Kontsevich 等作者。 然而，关于形式准同构空间上的这种作用，还存在几个悬而未决的问题，即，它在某种同伦意义上是传递的还是自由的。我希望研究这些问题能够加深我们对量子化的代数几何和动机方面的理解。