On the obstructed Lagrangian Floer theory

TL;DR

This paper investigates cohomology theories related to obstructed Lagrangian Floer theory, demonstrating their invariance and properties, and providing examples of vanishing and non-vanishing homologies in obstructed cases.

Contribution

It introduces well-defined cohomology theories for obstructed Lagrangian Floer objects and analyzes how obstructions affect their homological algebra, with explicit computations and examples.

Findings

Chevalley-Eilenberg Floer homology vanishes for obstructed cases with non-trivial obstructions.

Cyclic Floer homology can be non-vanishing even in obstructed cases.

The existence of $m_0$ influences the homological properties of these theories.

Abstract

Lagrangian Floer homology in a general case has been constructed by Fukaya, Oh, Ohta and Ono, where they construct an $\AI$-algebra or an $\AI$-bimodule from Lagrangian submanifolds, and studied the obstructions and deformation theories. But for obstructed Lagrangian submanifolds, standard Lagrangian Floer homology can not be defined. We explore several well-known cohomology theories on these $\AI$-objects and explore their properties, which are well-defined and invariant even in the obstructed cases. These are Hochschild and cyclic homology of an $\AI$-objects and Chevalley-Eilenberg or cyclic Chevalley-Eilenberg homology of their underlying $\LI$ objects. We explain how the existence of $m_0$ effects the usual homological algebra of these homology theories. We also provide some computations. We show that for an obstructed $\AI$-algebra with a non-trivial primary obstruction, Chevalley-Eilenberg Floer homology vanishes, whose proof is inspired by the comparison with cluster homology theory of Lagrangian submanifolds by Cornea and Lalonde. In contrast, we also provide an example of an obstructed case whose cyclic Floer homology is non-vanishing.