# A compactness theorem for stable flat $SL(2,\mathbb{C})$ connections on $3$-folds

## TL;DR

This paper establishes a compactness theorem for stable flat $SL(2,b{C})$ connections on 3-manifolds with non-degenerate flat $SU(2)$-connections, and shows the moduli space can be disconnected.

## Contribution

It proves a Uhlenbeck-type compactness theorem for stable flat $SL(2,b{C})$ connections and demonstrates the moduli space's disconnectedness under certain conditions.

## Key findings

- Proved a compactness theorem for stable flat $SL(2,b{C})$ connections.
- Showed the moduli space of these connections can be disconnected.
- Extended previous results to include $L^2$-bounded curvature connections.

## Abstract

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.