On the L2-Hodge theory of Landau-Ginzburg models

Abstract

Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus. We introduce the notion of f-twisted Sobolev spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property via L2-Hodge theoretical methods when f satisfies an asymptotic condition of strongly ellipticity. This leads to a Frobenius manifold via the Barannikov-Kontsevich construction, unifying the Landau-Ginzburg and Calabi-Yau geometry. Our construction can be viewed as a generalization of K.Saito's higher residue and primitive form theory for isolated singularities. As an application, we construct Frobenius manifolds for orbifold Landau-Ginzburg B-models which admit crepant resolutions.

41 pages, comments are welcome

References in corpus (1)

• Schrödinger equations, deformation theory and $t{t}^{\ast }$-geometry

Cited by in corpus (2)

• Unfolding of Orbifold LG B-Models: A Case Study
• Witten deformation for noncompact manifolds with bounded geometry