New York Journal of Mathematics
Volume 19 (2013) 305-342


Andrei Căldăraru and Junwu Tu

Curved A algebras and Landau-Ginzburg models

view    print

Published: June 12, 2013
Keywords: Curved algebra, Hochschild homology, matrix factorizations
Subject: Primary 18E30; secondary 14B05

We study the Hochschild (co)homology of curved A 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.


This material is based upon work supported by the National Science Foundation under Grant No. DMS-0901224.

Author information

Mathematics Department, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI 53706-1388, USA