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This expository paper details the theory of rank one Higgs bundles over aclosed Riemann surface X and their relationship to representations of thefundamental group of X. We construct an equivalence between the deformationtheories of flat connections and Higgs pairs. This provides an identificationof moduli spaces arising in different contexts. The moduli spaces are real Liegroups. From each context arises a complex structure, and the different complexstructures define a hyper-Kaehlerstructure. The twistor space, real forms, andvarious group actions are computed explicitly in terms of the Jacobian of X. Wedescribe the moduli spaces and their geometry in terms of the Riemann periodmatrix of X.  This is the simplest case of the theory developed by Hitchin, Simpson andothers. We emphasize its formal aspects that generalize to higher rank Higgsbundles over higher dimensional Kaehler manifolds.

Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces