C-Graded Vertex Algebras and their Representations

In this thesis we consider two related classes of vertex algebras. The first class we consider consists of objects called C-graded vertex algebras. These are vertex algebras with additional structure that allows for the construction of a Zhu algebra with a sufficiently well-behaved representation theory. This additional structure is minimal in the sense that it is necessary for the construction of the Zhu algebra. Given a C-graded vertex algebra, we provide a construction of the Zhu algebra and a pair of functors which are inverse bijections between the appropriate module categories.The second class we consider arises from considering conformal deformations of vertex operator algebras. These structures are called pseudo vertex operator algebras, and their main distinguishing feature is that the operator L(0) is not assumed to be semi-simple and is permitted to have complex eigenvalues. Similar theories have been studied: In the context of logarithmic conformal field theory, for example, L(0) is not required to be semi-simple on modules. Here, we extend that notion to allow L(0) to be non semi-simple on V itself. We show how to construct a family of pseudo vertex operator algebras from a given vertex operator algebra, and we prove that all such pseudo vertex operator algebras are C-graded vertex algebras. We then prove that every pseudo vertex operator algebra obtained via conformal deformation of a lattice vertex operator algebra is regular, which means that the category of admissible modules is semi-simple.