A Brief Survey of Higgs Bundles

TL;DR

This paper provides a concise overview of Higgs bundles, their historical development, mathematical structure, and significance across various fields like geometry, physics, and representation theory.

Contribution

It offers an introductory survey of Higgs bundles, highlighting their construction, historical context, and relevance in multiple research areas.

Findings

Introduction to Higgs bundle theory

Brief construction of the moduli space

Overview of applications in physics and mathematics

Abstract

Considering a compact Riemann surface of genus greater than two, a Higgs~bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin's work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as "Higgs fields" beacuse in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name "Higgs bundle" for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief construction of the moduli space of Higgs bundles.