Vertex Operator Algebras and Deformation Quantization of Poisson Algebras

对于顶点算子代数V，我们将空间PA(V)（V的子空间）关联起来，并根据PA(V)上的形式参数h定义了一个顶点算子的乘积*_{h}。从一些已知的结果可以得出，PA(V)是顶点算子代数V的生成子空间，具有与经典PBW定理相似的生成性质，并且PA(V)具有泊松代数结构。我们的第一个目标是证明（PA(V),*_{h}）是泊松代数PA(V)的*-变形，并证明（PA(V),*_{h}）唯一地决定了V上的顶点算子代数结构。我们的第二个目标是证明对于满足一定条件的具有*-变形结构*_{h}的给定抽象泊松代数P，存在一个顶点算子代数V，使得（PA(V),*_{h}）与（P,*_{h}）同构。已知*-变形具有几何根源。因此，我们的第三个目标是让顶点算子代数V使用上面的链接找到具有*-积结构的几何对象G(V)，该结构通常与（PA(V),*_{h}）相关。这些代数是从物理学中量子场论的研究中产生的。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，人们可能会对物理学的某些方面有新的认识。

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.