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

TL;DR

This paper proves a compactness theorem for $SO(3)$ anti-self-dual instantons with translation symmetry, advancing the understanding of their limiting behavior and contributing to the Atiyah-Floer conjecture.

Contribution

It establishes a Gromov-Uhlenbeck type compactness theorem for $SO(3)$ instantons on manifolds with cylindrical ends, including singular limits with instanton and holomorphic curve components.

Findings

Sequences of instantons with bounded energy have convergent subsequences.

Limits may include both instanton and holomorphic curve components.

This work is a foundational step towards the Fukaya's bounding cochain in the Atiyah-Floer conjecture.

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.