# Advanced topics in gauge theory: mathematics and physics of Higgs bundles

## TL;DR

This paper provides an overview of Higgs bundles, their properties, the Hitchin fibration, and the role of branes in the moduli space, connecting complex geometry with physical theories.

## Contribution

It offers a comprehensive introduction to Higgs bundles, their geometric structures, and the interplay between mathematical and physical perspectives, including branes and moduli space analysis.

## Key findings

- Detailed explanation of Higgs bundles and their properties
- Analysis of the Hitchin fibration and its applications
- Study of subspaces (branes) and their relations in the moduli space

## Abstract

These notes have been prepared as reading material for the mini-course given by the author at the "2019 Graduate Summer School" at Park City Mathematics Institute - Institute for Advanced Study. We begin by introducing Higgs bundles and their main properties (Lecture 1), and then we discuss the Hitchin fibration and its different uses (Lecture 2). The second half of the course is dedicated to studying different types of subspaces (branes) of the moduli space of complex Higgs bundles, their appearances in terms of flat connections and representations (Lecture 3), as well as correspondences between them (Lecture 4).