The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles

The following sections are included:

• Contents
• Foreword
• Preface

This chapter collects the notes of the lectures given on that subject during the introductory school of the program “The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles” (7 July - 29 August 2014). The main purpose is to state the correspondence between flat bundles and Higgs bundles and some of its features.

These notes provide an introduction to Higgs bundles for complex and real Lie groups, as well as a description of their associated spectral data.

This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed.

This chapter is based in part on lecture notes prepared for the summer school “The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles” at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a “bootstrap” method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure’s GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure’s moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin’s criteria.