Hitchin's projectively flat connection and the moduli space of Higgs bundles

Abstract

In this thesis we investigate the geometric quantization of moduli spaces of vector bundles over compact Riemann surfaces. In particular we will recall the geometric quantization of the moduli space of stable holomorphic vector bundles carried out by Hitchin, and study the generalisation of this problem for the moduli space of stable holomorphic Higgs bundles. The geometric quantization of Higgs moduli spaces presents new difficulties, since these moduli spaces are non-compact. However, they come with natural C+ actions, and this has implications for the geometric quantization: the quantum spaces for the Higgs moduli spaces split into finite-dimensional weight spaces for the C+ action, which can be identified with spaces of sections of certain bundles over the compact stable bundle moduli space. In the first part of this thesis, we review necessary background in differential geometry. Chapter 1 reviews the standard theory of connections on smooth vector bundles. Chapter 2 serves as an introduction to symplectic geometry, symplectic quotients, and their relationship to geometric invariant theory. Chapter 3 reviews the fundamental ideas in complex di erential geometry, and in particular in Kahler geometry, as well as the basic theory of holomorphic vector bundles required later. In the second part of this thesis, we introduce the moduli spaces of stable bundles and Higgs bundles, formally defining them as infinite-dimensional Kahler quotients. In Chapter 4 the stable bundle space is considered, and we review its important properties and interpretations. Chapter 5 concerns the Higgs bundle moduli space and the ways it generalises and compares to the stable bundle moduli space. In the third and final part of this thesis, we recall the process of geometric quantization via Kahler polarisations, and apply it to the moduli space of Higgs bundles. In Chapter 6 we define geometric quantization, and state a theorem of Andersen on the existence of Hitchin connections for compact symplectic manifolds. In Chapter 7 we geometrically quantize the Higgs line bundle moduli space, and investigate the difficulties of generalising the techniques used to the higher rank spaces. In particular we construct a projectively at Hitchin connection on the bundle of quantum spaces over Teichmuller space. As far as the author is aware, this is an original result.