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We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and some non-monotone Lagrangian torus fibers.

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 ChevalleyEilenberg or cyclic ChevalleyEilenberg 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 m 0, the standard homological algebra does not work and we develop analogous homological algebra over Novikov fields. We provide computations of these homology