To 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.
