Vertex Operator Algebras and Deformation Quantization of Poisson Algebras

9970496我们把一个空间PA(V)(V的一个子空间)联系到一个顶点算子代数V上，并根据PA(V)上的形参h定义了一个乘积*_{h}。证明了PA(V)是顶点算子代数V的生成子空间，其生成性类似于经典的PBW定理，并且PA(V)具有Poisson代数结构。我们的第一个目标是证明(PA(V)，*{h})是Poisson代数PA(V)的*-变形，并证明(PA(V)，*{h})唯一地决定V上的顶点算子代数结构。第二个目标是证明对给定的具有满足一定条件的*-变形结构*{h}的抽象Poisson代数P，存在一个顶点算子代数V使得(PA(V)，*{h})与(P，*{h})同构。我们知道，*-变形有一个几何起源。因此，我们的第三个目标是建立一个顶点算符代数V，利用上面的链接找到一个具有*-乘积结构的几何对象G(V)，它与(PA(V)，*{h})是典型相关的。这些代数源于物理学中量子场论的研究。与数论等其他领域的联系也存在。通过确定这些代数对象的性质，可能会对物理学的某些方面有新的见解。

9970496To a vertex operator algebra V we associate a space PA(V)(a subspace of V) and define a product *_{h} depending on a formal parameter h on PA(V) in terms of vertex operators. It follows from some known results that PA(V) is a generating subspace of the vertex operator algebra V with a spanning property similar to the one in the classical PBW theorem and that PA(V) has a Poisson algebra structure. Our first goal is to prove that (PA(V),*_{h}) is a *-deformation of the Poisson algebra PA(V) and to prove that (PA(V),*_{h}) uniquely determines the vertex operator algebra structure on V. Our second goal is to prove that for a given abstract Poisson algebra P equipped with a *-deformation structure *_{h}satisfying a certain condition, there exists a vertex operator algebra V such that (PA(V),*_{h}) is isomorphic to (P,*_{h}). It is known that *-deformation has a geometric origin. So our third goal is for a vertex operator algebra V to use the above link to find a geometric object G(V) with a *-product structure, which is canonically related to (PA(V),*_{h}).These algebras arise from the study of quantum field theory in physics. Connections with other areas like number theory also exist. By determining the properties of these algebraic objects new insights into certain aspects of physics might be expected.