  We study the Hochschild homology and cohomology of curved A-infinity algebras
that arise in the study of Landau-Ginzburg (LG) models in physics. We show that
the ordinary Hochschild homology and cohomology of these algebras vanish. To
correct this we introduce modified 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 particular 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.
