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

Abstract

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

arXiv comment71 pages, 4 figures. v2. corrected a mistake in the previous version which falsely claimed a diameter estimate. v3. shortened some arguments. v4. more details added as suggested by the referee. accepted by Advances in Mathematics