# Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

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*William M. Goldman; Eugene Z. Xia*

This expository article details the theory of
rank one Higgs bundles over a closed Riemann surface \(X\) and
their relation to representations of the fundamental group of
\(X\). The authors construct an equivalence between the
deformation theories of flat connections and Higgs pairs. This
provides an identification of moduli spaces arising in different
contexts. The moduli spaces are real Lie groups. From each context
arises a complex structure, and the different complex structures
define a hyperkähler structure. The twistor space, real forms,
and various group actions are computed explicitly in terms of the
Jacobian of \(X\). The authors describe the moduli spaces and
their geometry in terms of the Riemann period matrix of
\(X\).

This is the simplest case of the theory developed
by Hitchin, Simpson and others. The authors emphasize its formal
aspects that generalize to higher rank Higgs bundles over higher
dimensional Kähler manifolds.

#### Table of Contents

# Table of Contents

## Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

- Contents v6 free
- Introduction 110 free
- 1. Equivalences of deformation theories 413 free
- 2. The Betti and de Rham deformation theories and their moduli spaces 514
- 3. The Dolbeault groupoid 1726
- 4. Equivalence of de Rham and Dolbeault groupoids 2534
- 5. Hyperkahler geometry on the moduli space 3746
- 6. The twistor space 4352
- 6.1. The complex projective line 4352
- 6.2. The twistor space as a smooth vector bundle 4857
- 6.3. A holomorphic atlas for the twistor space 4958
- 6.4. The twistor lines 5160
- 6.5. The real structure on the twistor space 5261
- 6.6. Symplectic geometry of the twistor space 5362
- 6.7. The lattice quotient 5564
- 6.8. Functions and flows 5665

- 7. The moduli space and the Riemann period matrix 5968
- Bibliography 6776