Logarithmic Parafermion Vertex Operator Algebras

Abstract

This thesis is about the theory of vertex operator algebras and their representations. Its main results provide new examples of logarithmic C2-cofinite vertex operator algebras. These include closure of the characters under modular transformations with explicit determination of the modular coefficients for an infinite family of parafermionic vertex operator algebras and proofs of C2-cofiniteness for a few specific levels including for three new cases. The rest of the work presented in this thesis provides a new comprehension of the notoriously difficult proof of the Kac-Wakimoto character formula, but also categorical results and tools to study certain vertex operator algebras' categories involving infinite direct sums.