The Complex Gradient Flow Equation and Seidel's Spectral Sequence

TL;DR

This paper constructs Fukaya-Seidel categories using the complex gradient flow equation, offers an alternative proof of Seidel's spectral sequence, and explores potential generalizations to infinite-dimensional cases.

Contribution

It introduces a new approach to Fukaya-Seidel categories via complex gradient flows and provides an alternative proof of Seidel's spectral sequence, linking it to geometric filtrations.

Findings

Established a geometric filtration on Floer cochain complexes.

Identified the spectral sequence with Seidel's original one.

Potential for generalization to infinite-dimensional models.

Abstract

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