Curved A ∞ algebras and Landau-Ginzburg models

We study the Hochschild homology and cohomology of curved A∞ algebras that arise in the study of Landau-Ginzburg (LG) models in physics. We show that the ordinary Hochschild homology and coho-mology of these algebras vanish. To correct this we introduce modi-fied versions of these theories, Borel-Moore Hochschild homology and compactly supported Hochschild cohomology. For LG models the new invariants yield the answer predicted by physics, shifts of the Jacobian ring. We also study the relationship between graded LG models and the geometry of hypersurfaces. We prove that Orlov’s derived equivalence descends from an equivalence at the differential graded level, so in par-ticular the CY/LG correspondence is a dg equivalence. This leads us to study the equivariant Hochschild homology of orbifold LG models. The results we get can be seen as noncommutative analogues of the Lefschetz hyperplane and Griffiths transversality theorems. 1