This is a part of an ongoing project with Aliakbar Daemi studying Instanton Floer homology for 3 manifold with boundary and Atiyah-Floer conjecture. We need to study certain moduli space which mixed up Yang-Mills equation and pseudo-holomorphic curve equation. In the pseudo-holomorphic curve equation part we need virtual fundaental chain technique to work out transversality issue in a reasonable level of genelarity. In gauge theory part people usually use Uhlenbeck compactification, which is not described by perfect obstruction theory.
In this talk I will explain how we adapt the framework of virtual fundamental chain
so that it can include Uhlenbeck compactification.
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