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Let N be a closed four dimensional manifold which admits a self-indexingMorse function f with only 3 critical values 0,2,4, and a unique maximum andminimum. Let g be a Riemannian metric on N such that (f,g) is Morse-Smale. Weconstruct from (N,f,g) a certain six dimensional exact symplectic manifold M,together with some exact Lagrangian spheres V_4, V_2^j, V_0 in M, j=1,...,k.These spheres correspond to the critical points x_4, x_2^j, x_0 of f, where thesubscript indicates the Morse index. (In a companion paper we explain how (M,V_4,{V_2^j},V_0) is a model for the regular fiber and vanishing spheres of thecomplexification of f, viewed as a Lefschetz fibration on the disk cotangentbundle D(T^*N).) Our main result is a computation of the Lagrangian Floerhomology groups HF(V_4,V_2^j), HF(V_2^j,V_0), HF(V_4,V_0) and the triangleproduct mu_2: HF(V_4,V_2^j) \otimes HF(V_2^j,V_0) --> HF(V_4,V_0). The outcomeis that the directed Donaldson-Fukaya category of (M,V_4,{V_2^j},V_0) isisomorphic to the flow category of (N,f,g).

Complexifications of Morse functions and the directed Donaldson-Fukaya category