# Torsion Subgroups of Jacobians Acting on Moduli Spaces of Vector Bundles

By Frank Nasser
PhD Dissertations
June 2005
Abstract:

This thesis is associated to the area of mathematics known as topological quantum field theory.

In ultra-brief, the purpose of topological quantum field theory is to build invariants of three manifolds by constructing certain functors from a cobordism category of three manifolds to vector spaces. A well-known way to do this is to construct so-called modular functors, associating vector spaces to closed, oriented surfaces.

The thesis is concerned with an aspect of the gauge theoretic approach to the construction of modular functors. -More specifically, with a natural structure of the TQFT vector spaces as representations of certain finite groups.

However, whilst being motivated and inspired by both gauge theory and topological quantum field theory, the main body of work in the thesis is algebraic geometric by nature.

In order to define the representations, a thorough understanding is needed of the natural action of torsion subgroups of the Jacobian variety of a Riemann surface on the moduli spaces of semistable vector bundles on that surface.

In particular, the geometry of the fixed point varieties of the action is studied, with emphasis on intersection properties. The outcome of this study is used to define certain groups of lifts, acting on the determinant line bundles on the moduli spaces.