Components of spaces of representations and stable triples

Details Components of spaces of representations and stable triples Components of spaces of representations and stable triples

1999-04-21

arxiv.org/abs/math/9904114v2

Topology 40 (2001), 823-850

Mathematics

Algebraic Geometry

28 pages

Scientific paper

We consider the moduli spaces of representations of the fundamental group of a surface of genus g greater than 2 in the Lie groups SU(2,2) and Sp(4,R). It is well known that there is a characteristic number of such a representation, whose absolute value is less than or equal to 2g-2. This allows one to write the moduli space as a union of subspaces indexed by the characteristic number, each of which is a union of connected components. The main result of this paper is that the subspaces with characteristic number plus or minus 2g-2 are connected in the case of representations in SU(2,2), while they break up into 2^{2g+1}+2g-3 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 Garcia-Prada.

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