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Non-Abelian Hodge Theory and Moduli Spaces of Higgs Bundles

Guillermo Gallego

TL;DR

The paper surveys non-abelian Hodge theory and the moduli spaces of Higgs bundles on compact Riemann surfaces, unifying algebraic, differential, and topological perspectives through the hyperkähler Hitchin moduli space. It develops the moduli theory for vector and Higgs bundles, presents the Crux of the non-abelian Hodge correspondences (Betti/de Rham/Dolbeault) via the Hitchin system, and analyzes the Hitchin fibration, spectral data, and dualities including SYZ and topological mirror symmetry. It also covers real-structure extensions, deformations, and the rank-two computations that verify topological mirror symmetry, illustrating methods like Morse theory and spectral curves. The work emphasizes the geometric, topological, and representation-theoretic aspects of NAHT and places it in the context of Langlands duality and geometric representation theory, highlighting the interplay between Hitchin fibrations, spectral curves, and moduli stacks. Overall, the text provides a cohesive, expository framework connecting moduli of bundles, flat connections, Higgs fields, and their rich geometric structures with explicit rank-two calculations that exemplify the theory’s predictive power and connections to mirror symmetry.

Abstract

This paper provides an introduction to non-abelian Hodge theory and moduli spaces of Higgs bundles on compact Riemann surfaces. We develop the moduli theory of vector bundles and Higgs bundles, establish the main correspondences of non-abelian Hodge theory, and interpret them through the hyperkähler structure on the Hitchin moduli space. We study the Hitchin fibration and its geometric properties, including SYZ mirror symmetry and topological mirror symmetry for type $\mathsf{A}$ Hitchin systems. As an illustration, we compute the Poincaré polynomial of the rank 2 moduli space and verify topological mirror symmetry in this case.

Non-Abelian Hodge Theory and Moduli Spaces of Higgs Bundles

TL;DR

The paper surveys non-abelian Hodge theory and the moduli spaces of Higgs bundles on compact Riemann surfaces, unifying algebraic, differential, and topological perspectives through the hyperkähler Hitchin moduli space. It develops the moduli theory for vector and Higgs bundles, presents the Crux of the non-abelian Hodge correspondences (Betti/de Rham/Dolbeault) via the Hitchin system, and analyzes the Hitchin fibration, spectral data, and dualities including SYZ and topological mirror symmetry. It also covers real-structure extensions, deformations, and the rank-two computations that verify topological mirror symmetry, illustrating methods like Morse theory and spectral curves. The work emphasizes the geometric, topological, and representation-theoretic aspects of NAHT and places it in the context of Langlands duality and geometric representation theory, highlighting the interplay between Hitchin fibrations, spectral curves, and moduli stacks. Overall, the text provides a cohesive, expository framework connecting moduli of bundles, flat connections, Higgs fields, and their rich geometric structures with explicit rank-two calculations that exemplify the theory’s predictive power and connections to mirror symmetry.

Abstract

This paper provides an introduction to non-abelian Hodge theory and moduli spaces of Higgs bundles on compact Riemann surfaces. We develop the moduli theory of vector bundles and Higgs bundles, establish the main correspondences of non-abelian Hodge theory, and interpret them through the hyperkähler structure on the Hitchin moduli space. We study the Hitchin fibration and its geometric properties, including SYZ mirror symmetry and topological mirror symmetry for type Hitchin systems. As an illustration, we compute the Poincaré polynomial of the rank 2 moduli space and verify topological mirror symmetry in this case.
Paper Structure (53 sections, 42 theorems, 367 equations, 1 figure)

This paper contains 53 sections, 42 theorems, 367 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Abelian Hodge theory
  3. Non-abelian Hodge theory: A short history of the topic
  4. Recent developments
  5. About this paper
  6. Organization of the paper
  7. Acknowledgements
  8. A primer on vector bundles and connections
  9. Vector bundles in different categories
  10. Connections and curvature
  11. Flat bundles and local systems
  12. Holomorphic structures as Dolbeault operators
  13. Holomorphic connections
  14. Hermitian metrics and the Chern correspondence
  15. Principal $G$-bundles
  16. ...and 38 more sections

Key Result

Theorem 2.12

Let $(E,D)$ be a flat bundle. Suppose that $E$ is determined by a cocycle $(g_{UV})$. Then there exists a $0$-coboundary $(f_U)$ such that the functions $g'_{UV}=(f_V)^{-1} g_{UV} f_U$ are locally constant. The corresponding local system $E'$ determined by $(g'_{UV})$ is called the holonomy local sy

Figures (1)

  • Figure 1: The spectral curve

Theorems & Definitions (98)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 88 more