Vertex operator algebras: Finite-dimensional cases and conformal blocks

Abstract

Vertex operator algebras are algebraic objects analogous to both commutative associative algebras with identity and Lie algebras. They provide a way of rigorously constructing a particular family of quantum field theories called rational conformal field theories. In this thesis we construct the simplest class of examples of vertex operator algebras, namely the finite dimensional ones, and prove basic results on modules of these vertex operator algebras and spaces of conformal blocks associated to smooth projective curves. We also construct the vertex operator algebra associated with the $\mathfrak{sl}_2(\mathbb{C})$ WZW model in the non-critical case. When combined with the FRS theorem for rational conformal field theories, vertex operator algebra theory can be used to rigorously construct one of the simplest examples of holographic duality: the Chern-Simons-WZW model correspondence.