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）联系起来，并根据顶点算子定义了依赖于PA（V）上的形式参数h的乘积 *_{h}。 从已有的结果可以看出，PA（V）是顶点算子代数V的生成子空间，其生成性质类似于经典PBW定理中的生成性质，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）的一个 *-变形，并证明（PA（V），* h}）唯一地决定V上的顶点算子代数结构。我们的第二个目标是证明对于给定的抽象Poisson代数P，它具有一个满足一定条件的 *-变形结构 * h}，存在顶点算子代数V使得（PA（V），* h）同构于（P，* h）. 已知 *-形变有一个几何起源。 所以我们的第三个目标是使顶点算子代数V利用上述联系找到一个具有 *-积结构的几何对象G（V），它与（PA（V），* h}）正则相关。与其他领域的联系，如数论也存在。 通过确定这些代数对象的属性，可以期望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.