Memoirs of the American Mathematical Society 2008; 69 pp; softcover Volume: 193 ISBN10: 082184136X ISBN13: 9780821841365 List Price: US$61 Individual Members: US$36.60 Institutional Members: US$48.80 Order Code: MEMO/193/904
 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  Introduction
 Equivalences of deformation theories
 The Betti and de Rham deformation theories and their moduli spaces
 The Dolbeault groupoid
 Equivalence of de Rham and Dolbeault groupoids
 Hyperkähler geometry on the moduli space
 The twistor space
 The moduli space and the Riemann period matrix
 Bibliography
