  This expository paper details the theory of rank one Higgs bundles over a
closed Riemann surface X and their relationship to representations of the
fundamental group of X. We 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 hyper-Kaehlerstructure. The twistor space, real forms, and
various group actions are computed explicitly in terms of the Jacobian of X. We
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. We emphasize its formal aspects that generalize to higher rank Higgs
bundles over higher dimensional Kaehler manifolds.
