Algebraic Structures Related to Quantum Field Theory

目前的建议涉及几个代数结构的二维共形场论，以及与量子场论在时空维度高于2的高维类似物。研究的结构是：顶点代数，（非线性）李共形代数，李伪代数，和顶点（李）代数在更高的维度。提出了三个主要项目。第一个是对一类具有三个生成元的非线性共形李代数进行分类，这些共形李代数可以看作仿射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，扩展了PI和合作者以前解决W型和S型的工作。这个项目是密切相关的表示理论的李-嘉当代数的向量场，它涉及到伪代数版本的哈密尔顿和接触德拉姆复合。第三个项目进一步发展了N.M. Nikolov的长期目标是构建导致公理化量子场论的非平凡模型的例子。PI将继续他的调查顶点李代数在更高的维度介绍了他和尼科洛夫。李群是以19世纪世纪挪威数学家Sophus Lie的名字命名的，它提供了连续对称概念的数学描述，在数学和物理学中扮演着重要的角色。相关的对象，李代数，描述无穷小变换。PI研究无限维李代数和相关的代数结构，它首先出现在量子场论的代数方法中。本研究的目标是进一步发展这种代数的数学理论，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，从而加深我们对自然的理解。

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.