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Motivated by the Atiyah-Floer conjecture, we consider $SO(3)$ Santi-self-dualinstantons on the product of the real line and a three-manifold withcylindrical end. We prove a Gromov-Uhlenbeck type compactness theorem, namely,any sequence of such instantons with uniform energy bound has a subsequenceconverging to a type of singular objects which may have both instanton andholomorphic curve components. This result is the first step towardsconstructing a natural bounding cochain proposed by Fukaya for the $SO(3)$Atiyah-Floer conjecture.

A Compactness Theorem for $SO(3)$ Anti-Self-Dual Equation with Translation Symmetry