Components of spaces of representations and stable triples

We consider the moduli spaces of representations of the fundamental group of a surface of genus g⩾2 in the Lie groups SU(2, 2) and Sp(4,R). It is well known that there is a characteristic number, d, of such a representation, satisfying the inequality |d|⩽2g−2. This allows one to write the moduli space as a union of subspaces indexed by d, each of which is a union of connected components. The main result of this paper is that the subspaces corresponding to d=±(2g−2) are connected in the case of representations in SU(2, 2), while they break up into 3·22g+2g−4 connected components in the case of representations in Sp(4, R). We obtain our results using the interpretation of the moduli space of representations as a moduli space of Higgs bundles, and an important step is an identification of certain subspaces as moduli spaces of stable triples, as studied by Bradlow and Garcı́a-Prada.