On the obstructed Lagrangian Floer theory

Lagrangian Floer homology in a general case has been constructed by Fukaya, Oh, Ohta and Ono, where they construct an A∞-algebra or an A∞-bimodule from Lagrangian submanifolds. They developed obstruction and deformation theories of the Lagrangian Floer homology theory. But for obstructed Lagrangian submanifolds, the standard Lagrangian Floer homology cannot be defined.We explore several well-known homology theories on these A∞-objects, which are Hochschild and cyclic homology for an A∞-objects and Chevalley-Eilenberg or cyclic Chevalley-Eilenberg homology for their underlying L∞-objects. We show that these homology theories are well-defined and invariant even in the obstructed cases. Due to the existence of m0, the standard homological algebra does not work and we develop analogous homological algebra over Novikov fields.We provide computations of these homology theories in some cases: We show that for an obstructed A∞-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.