Curved A-infinity algebras and Landau-Ginzburg models

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.

💡 Research Summary

The paper investigates Hochschild homology and cohomology for curved A‑infinity algebras that naturally arise in Landau‑Ginzburg (LG) models, a class of supersymmetric quantum field theories. The authors begin by recalling that a curved A‑infinity algebra is an A‑infinity algebra equipped with a degree‑zero curvature element m₀, which modifies the usual A‑infinity relations and forces the differential m₁ to interact with m₀. When one constructs the ordinary Hochschild chain complex for such an algebra, the curvature term annihilates every cycle, leading to the striking result that both ordinary Hochschild homology and cohomology vanish identically. This vanishing contradicts physical expectations, because in LG theory the space of B‑type topological observables is predicted to be the Jacobian ring of the superpotential W, i.e. k