The Hitchin fibration is one of the central structures in modern geometry. It connects:

• Higgs bundles
• integrable systems
• non-abelian Hodge theory
• geometric Langlands
• mirror symmetry

This page collects a recommended path to learn the subject, with references.

The starting point of the whole subject.

Nigel Hitchin (1987) “Stable bundles and integrable systems”

Introduces:

• Higgs bundles
• the Hitchin map
• the integrable system structure
• the Morse function \(f = |\phi|^2\)

Link: https://doi.org/10.1215/S0012-7094-87-05408-1

PDF: https://people.maths.ox.ac.uk/hitchin/hitchin87.pdf

Key ideas to focus on:

• definition of Higgs bundle
• Hitchin base
• spectral curves
• Morse theory on the moduli space

This explains the spectral correspondence.

Beauville – Narasimhan – Ramanan

“Spectral curves and the generalized theta divisor”

Link: https://math.univ-cotedazur.fr/~beauvill/pubs/bnr.pdf

The main result:

(E, φ) ↔ line bundle on spectral curve

Consequences:

• Hitchin fibers become Jacobians
• Hitchin system becomes an algebraically completely integrable system.

Carlos Simpson (1992) “Higgs bundles and local systems”

Link: https://www.numdam.org/item/PMIHES_1992__75__5_0.pdf

Explains the correspondence:

Higgs bundles ↕ flat connections ↕ representations of π₁

Important features:

• harmonic metrics
• C* action on Higgs moduli
• fixed points (VHS)
• Morse theory of Hitchin function

Gérard Laumon

"Un analogue global du cône nilpotent"

Main result:

• the global nilpotent cone is a Lagrangian substack of the cotangent stack of Bun.

Key ideas:

• stratification by nilpotent type
• relation to Harder–Narasimhan strata
• geometry of Hitchin fiber over 0

Hausel – Thaddeus

“Mirror symmetry, Langlands duality, and the Hitchin system”

Link: https://arxiv.org/abs/math/0205236

Very good conceptual overview of:

• Hitchin integrable systems
• mirror symmetry
• Langlands duality
• topology of moduli spaces

Ngô Bao Châu

Work on the Hitchin fibration used to prove the Fundamental Lemma.

Example reference:

https://arxiv.org/pdf/0801.0446

Key themes:

• geometry of singular Hitchin fibers
• support theorem
• role of Hitchin fibration in geometric Langlands

Tamás Hausel – Hitchin systems

https://hausel.pages.ist.ac.at/wp-content/uploads/sites/229/2024/09/gths.pdf

Good for:

• topology of Higgs moduli
• Morse theory
• integrable system viewpoint

Pavel Etingof and Henry Liu, lecture at BIMSA

https://arxiv.org/pdf/2409.09505

Important themes when studying the Hitchin fibration:

• spectral curves
• integrable system structure
• C* action on Higgs moduli
• Morse theory of f = |φ|²
• nilpotent cone
• singular Hitchin fibers