The Extended Topological Quantum Field Theory Of The Fukaya Category In Yang-Mills Theory

Abstract

The Atiyah-Floer conjecture links symplectic topology and low-dimensional geometry. It claims that the Donaldson- Fukaya categories of Atiyah-Bott moduli spaces describe the behaviour of gauge-theoretic invariants of 3- and 4-manifolds under gluing operations. This claim can be formulated as the existence of an extended topological quantum field theory arising from Yang-Mills theory in dimensions 2, 3 and 4. More precisely, the conjecture claims the existence of a (‘natural’) isomorphism between the instanton Floer homology HFI (Y ) of a homology 3-sphere Y , and the Lagrangian intersection Floer homology HFL(L1 L2) of the two (generally immersed) Lagrangian submanifolds L1 L2 of the (symplectic) moduli spaceM of flat connections (over a Riemann surface ) arising by restriction from a Heegaard splitting Y = Y1 [ Y2 of Y along . AlthoughM is a monotone symplectic manifold (whenever it is smooth) because L1 and L2 are immersed, the Lagrangian intersection Floer homology may fail to exist due to the appearance of anomalies. Using the obstruction theory of Fukaya-Oh-Ohta-Ono, and its extension to the immersed case by Akaho-Joyce [AJ10], whenever L1 and L2 are unobstructed, suitable bounding cochains bL1 bL2 can be used to deform the boundary map for Lagrangian intersection Floer homology and hence define HFL((L1 bL1) (L2 bL2)). In his 2015 paper [Fuk15], Fukaya shows that this is indeed the case: the Lagrangians L1 L2 are unobstructed and moreover we have a canonical choice of bounding cochain. In this case, Fukaya claims THEOREM 0.1. (Fukaya, 2015) WheneverM is smooth, we have HFI (Y1 [ Y2) = HFL((L1 bL1) (L2 bL2)) This thesis explains the above statement, defining the groups on both sides of this isomorphism in the case where all the relevant moduli spaces are transversal. In this case, one obtains a considerable simplification of the obstruction theory of Fukaya-Oh-Ohta-Ono when one uses instead a de Rham model of cohomology. Finally, we discuss how Fukaya claims to prove the above statement, and the various directions in which this result might be taken.