Complexifications of Morse functions and the directed Donaldson-Fukaya category

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Let N be a closed four dimensional manifold which admits a self-indexing Morse function f with only 3 critical values 0,2,4, and a unique maximum and minimum. Let g be a Riemannian metric on N such that (f,g) is Morse-Smale. We construct from (N,f,g) a certain six dimensional exact symplectic manifold M, together with some exact Lagrangian spheres V₄, V₂ʲ, V₀ in M, j=1,...,k. These spheres correspond to the critical points x₄, x₂ʲ, x₀ of f, where the subscript indicates the Morse index. (In a companion paper we explain how (M, V₄,V₂ʲ,V₀) is a model for the regular fiber and vanishing spheres of the complexification of f, viewed as a Lefschetz fibration on the disk cotangent bundle D(T*N).) Our main result is a computation of the Lagrangian Floer homology groups HF(V₄,V₂ʲ), HF(V₂ʲ,V₀), HF(V₄,V₀) and the triangle product mu₂: HF(V₄,V₂ʲ) ⊗ HF(V₂ʲ,V₀) --> HF(V₄,V₀). The outcome is that the directed Donaldson-Fukaya category of (M,V₄,V₂ʲ,V₀) is isomorphic to the flow category of (N,f,g).