Cgraded vertex algebras and their representations
Abstract
In this thesis we consider two related classes of vertex algebras. The first class we consider consists of objects called Cgraded vertex algebras. These are vertex algebras with additional structure that allows for the construction of a Zhu algebra with a sufficiently wellbehaved representation theory. This additional structure is minimal in the sense that it is necessary for the construction of the Zhu algebra. Given a Cgraded 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 semisimple 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 semisimple on modules. Here, we extend that notion to allow L(0) to be non semisimple 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 Cgraded 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 semisimple.
 Publication:

Ph.D. Thesis
 Pub Date:
 2014
 Bibcode:
 2014PhDT.......103L
 Keywords:

 Mathematics;Theoretical Mathematics;Physics, Theory