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

2019/07/24 by Guangbo Xu, Xu, Guangbo
Mathematics · #57R58 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1907.10205

openalex publication_date 2019/07/24 · openalex created_date 2022/12/27 · openalex updated_date 2026/07/28

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.

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