  For a Landau Ginzburg space ([C^n/G],W), we construct the Witten's top Chern
classes as algebraic cycles via cosection localized virtual cycles in case all
sectors are narrow. We verify all axioms of such classes. We derive an explicit
formula of such classes in the free case. We prove that this construction is
equivalent to the prior constructions of Polishchuk-Vaintrob, of Chiodo and of
Fan-Jarvis-Ruan.
