Zhu reduction theory for vertex operator algebras on Riemann surfaces

Abstract

In this thesis we first develop a recursive relation for $n$-point functions for Vertex Operator Super Algebras (VOSAs) on a genus two Riemann surface constructed by sewing two tori. This relation is used to develop formal differential equations for $n$-point functions on a genus two surface, as well as for differential forms on this surface. We demonstrate the applications of this results for a well-known example of VOSA and compare them to existing results in the literature. In the second part, we develop a more general version of this identity for a Vertex Operator Algebra (VOA) on a general genus Riemann surface, using the Schottky uniformisation of a genus $g$ Riemann surface; we then develop some geometric theory for the results that arise. We also apply this results to well-known examples of VOAs to obtain general genus identities for objects such as differential forms on a Riemann surface.