A Gentle Introduction to the Non-Abelian Hodge Correspondence

TL;DR

This paper provides a comprehensive, pedagogical overview of the non-abelian Hodge correspondence, explaining its connections to algebra, geometry, and complex analysis, with emphasis on key concepts and applications.

Contribution

It offers an accessible introduction to the non-abelian Hodge correspondence, highlighting its foundational ideas, links to various mathematical theories, and recent developments.

Findings

Clarifies the relationship between fundamental group representations and holomorphic bundles.

Explains the role of Higgs bundles and harmonic bundles in the correspondence.

Discusses applications such as the Riemann--Hilbert correspondence and symplectic structures.

Abstract

We aim at giving a pedagogical introduction to the non-abelian Hodge correspondence, a bridge between algebra, geometric structures and complex geometry. The correspondence links representations of a fundamental group, the character variety, to the theory of holomorphic bundles. We focus on motivations, key ideas, links between the concepts and applications. Among others we discuss the Riemann--Hilbert correspondence, Goldman's symplectic structure via the Atiyah--Bott reduction, the Narasimhan--Seshadri theorem, Higgs bundles, harmonic bundles and hyperk\"ahler manifolds.