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Following the proposals of Donaldson-Thomas, Haydys and Gaiotto-Moore-Witten,we give a construction of Fukaya-Seidel categories for a suitable class ofMorse Landau-Ginzburg models using the complex gradient flow equation, whichhas the potential for generalization to some infinite dimensional examples.  In the course of this construction, we give an alternative proof to Seidel'sspectral sequence for Lagrangian Floer cohomology, which can be viewed as afinite dimensional model for a potential bordered monopole Floer theory. Thekey observation is that under a neck-stretching limit, this complex gradientflow equation produces a natural geometric filtration on the Floer cochaincomplex. The resulting spectral sequence is then identified with Seidel'soriginal one.

The Complex Gradient Flow Equation and Seidel's Spectral Sequence