Curved A-infinity algebras and Landau-Ginzburg models

TL;DR

This paper investigates curved A-infinity algebras in Landau-Ginzburg models, introduces modified Hochschild invariants to capture physics predictions, and establishes derived equivalences linking algebraic and geometric aspects of LG models.

Contribution

It introduces Borel-Moore Hochschild homology and compactly supported Hochschild cohomology for curved A-infinity algebras, aligning algebraic invariants with physical predictions and connecting to geometric dualities.

Findings

Modified Hochschild invariants match physics predictions.

Derived equivalence extends to dg level, confirming CY/LG correspondence.

Results generalize Lefschetz hyperplane and Griffiths theorems.

Abstract

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.