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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This is a report on a joint work with A. Daemi and M. Lipyanskiy.ÌýThe instanton Floer homology associate a graded vector spaceÌýto a 3 manifold with SO(3) bundle whose second Stiefel-Whitnely classÌýis nontrivial. The goal is to enhance it to the case of a 3 manifold with boundaryÌýso that SO(3) bundle is nontrivial on each connected componentÌýof the boundary. The invariant we obtain is an immersed Lagrangian submanifoldÌýof the moduli space of flat connections of 2 manifold,Ìýequipped with certain finite sum of its self intersection points,Ìýwhich 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 theÌýproject some more issues to clarify appears. I believe we now understandÌýhow to resolve those issues and want to explain some of them in this talk.