  We classify the connected components of the space of representations of the
fundamental group of a closed oriented surface of genus $\geq 2$ in
$Sp(4,{\mathbf R})$. We prove that this is equivalent to classifying the
connected components of the moduli space of representations (which consists of
conjugacy classes of reductive representations). We then continue the analysis
initiated by Gothen in math/9904114, relying on: (1) the relation between our
moduli space of representations and the moduli space of certain (polystable)
analogues of Higgs bundles (which follows essentially from work of Hitchin,
Simpson and Corlette) and (2) the existence of a proper function on the moduli
space whose set of local minima can be identified with the moduli space $N$ of
(polystable) quadratic bundles of rank 2; we finally deduce our result from the
classification of the connected components of $N$.
