  Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on
$Y$ are $non$-$degenerate$. In this article, we prove a Uhlenbeck-type
compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections
satisfying an $L^{2}$-bound for the real curvature. Combining the compactness
theorem and a previous result in \cite{Huang}, we prove that the moduli space
of the stable flat $SL(2,\mathbb{C})$ connections is disconnected under certain
technical assumptions.
