Algebraic Structures Related to Quantum Field Theory

本方案涉及由二维共形场论激发的几种代数结构，以及与时空维度高于2的量子场论相关的高维类似物。所研究的结构是：顶点代数、(非线性)李共形代数、李伪代数和高维顶点(李)代数。提出了三个主要项目。第一种方法是对某些具有三个生成元的非线性李共形代数进行分类，它们可以看作是仿射sl(2)的非线性变形。这将导致新的有趣的顶点代数的发现。第二个项目是对H型和K型单李伪代数的不可约表示进行分类，推广了PI和合作者解决了W型和S型的工作。这个项目与向量场的Lie-Cartan代数的表示理论密切相关，它涉及到哈密顿复形和联络De Rham复形的伪代数版本。第三个项目进一步发展了N.M.尼科洛夫之前介绍的高维顶点代数理论，长期目标是构建导致公理量子场论的非平凡模型的例子。PI将继续他和尼科洛夫介绍的关于高维顶点李代数的研究。本文提出的另一种方法是从高维共形顶点代数的一维限制和无限小保角变换的李代数的作用来重构。李群是以19世纪挪威数学家索菲斯·李的名字命名的，它提供了连续对称性概念的数学描述，在数学和物理学中发挥着重要作用。相关的对象，李代数，描述无限小的变换。PI研究无限维李代数和相关的代数结构，它们最早出现在量子场论的代数方法中。这项研究的目的是进一步发展此类代数的数学理论，同时发现新的具体例子，这将导致量子场论中新模型的构建，从而加深我们对本质的理解。

The present proposal deals with several algebraic structures motivated by two-dimensional conformal field theory, as well as with higher-dimensional analogs related to quantum field theory in space-time dimension higher than two. The structures investigated are: vertex algebras, (non-linear) Lie conformal algebras, Lie pseudoalgebras, and vertex (Lie) algebras in higher dimensions. Three main projects are proposed. The first is to classify certain non-linear Lie conformal algebras with three generators, which can be viewed as non-linear deformations of affine sl(2). This will lead to the discovery of new interesting vertex algebras. The second project is to classify the irreducible representations of simple Lie pseudoalgebras of type H and K, extending previous work of the PI and collaborators that settled types W and S. This project is closely related to the representation theory of Lie-Cartan algebras of vector fields, and it involves pseudoalgebra versions of the hamiltonian and contact de Rham complexes. The third project develops further the theory of vertex algebras in higher dimensions introduced previously by N.M. Nikolov, with the long-term goal of constructing examples that lead to nontrivial models of axiomatic quantum field theory. The PI will continue his investigation of the vertex Lie algebras in higher dimensions introduced by him and Nikolov. Another approach proposed here is to reconstruct a conformal vertex algebra in higher dimensions from its one-dimensional restriction and the action of the Lie algebra of infinitesimal conformal transformations.Lie groups, named after the 19th century Norwegian mathematician Sophus Lie, provide a mathematical description of the notion of continuous symmetry and play a prominent role in mathematics and physics. Related objects, Lie algebras, describe infinitesimal transformations. The PI investigates infinite-dimensional Lie algebras and related algebraic structures, which first appeared in an algebraic approach to quantum field theory. The goal of this research is to develop further the mathematical theory of such algebras and at the same time discover new concrete examples, which will lead to the construction of new models in quantum field theory and thus will deepen our understanding of the nature.