Homological Methods in Quantum Field Theory

计划重点研究量子场论和量子力学中与量子化过程有关的几个问题。这些问题包括：1)余弦代数的量子化；2) Batalin-Vilkovitski形式主义的数学精确表述；Grothendieck-Teichmueller群对各种形式态和量化态射的作用。前两个问题出现在量子场论中。由A. Beilinson和V. Drinfeld提出了余弦代数的量化问题；他们找到了一个非常不平凡的方法来解决这个问题在一个特殊的线性余弦括号的情况下。一般的量化问题要困难得多，用类似的方法是无法解决的。适当的工具可以是变形理论和在适当的变形复合体上引入附加结构。巴塔林-维尔科维茨基形式主义是具有复杂规范对称性的系统的量子化中最有力的工具之一。这种形式意味着路径积分的广泛使用，因此缺乏最重要的结构成分的数学意义，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导致发散。只有仔细分析巴塔林-维尔科维茨基的形式主义，包括d模理论和同态代数，才能让人们构建一个数学上有意义的理论。从Kontsevich形式定理可知，Grothendieck-Teichmueller群的作用既存在于李双代数的量子化函子集上，也存在于形式拟同构集上。这个引人入胜的主题是由V.德林菲尔德（V. Drinfeld）发起的，并由p . ettingf - d .进一步发展。Kazhdan， M. Kontsevich和其他作者。然而，关于这种对形式性准同构空间的作用，有几个悬而未决的问题，即，它在某种同主题意义上是及物的还是自由的。我希望研究这些问题能加深我们对量化的代数几何和动机方面的理解。

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.