SO(3) Instanton Floer homology as a 2-3 dimensional topological Field theory

SO(3) Instanton Floer homology as a 2-3 dimensional topological Field theory

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Kenji Fukaya, Stony Brook University

Zoom link:  https://princeton.zoom.us/j/99136657600

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This is a report on a joint work with A. Daemi and M. Lipyanskiy. The instanton Floer homology associate a graded vector space to a 3 manifold with SO(3) bundle whose second Stiefel-Whitnely class is nontrivial. The goal is to enhance it to the case of a 3 manifold with boundary so that SO(3) bundle is nontrivial on each connected component of the boundary. The invariant we obtain is an immersed Lagrangian submanifold of the moduli space of flat connections of 2 manifold, equipped with certain finite sum of its self intersection points, which is a bounding co-chain in the sense of FOOO and Akaho-Joince. The gluing formula for this topological field theory is certain version of Atiyah-Floer conjecture. We started this project more than 3-4 yours ago and while we are working out the project some more issues to clarify appears. I believe we now understand how to resolve those issues and want to explain some of them in this talk.