$L^2$-Hodge theoretic construction of Frobenius manifolds for Calabi-Yau smooth projective hypersurfaces

TL;DR

This paper introduces an $L^2$-Hodge theoretic method to construct Frobenius manifolds on Calabi-Yau hypersurfaces, linking it to existing constructions and providing a new geometric perspective.

Contribution

It presents a novel $L^2$-Hodge theoretic approach to build Frobenius manifolds for Calabi-Yau hypersurfaces, connecting it with Barannikov-Kontsevich's framework.

Findings

Constructed Frobenius manifolds using $L^2$-Hodge theory.

Established a comparison between the new and existing Frobenius structures.

Provided geometric insights into the Landau-Ginzburg model's role.

Abstract

We provide a new $L^2$-Hodge theoretic construction of a Frobenius manifold structure on the cohomology of a Calabi-Yau smooth projective hypersurface $V$, using Li-Wen's $L^2$-Hodge theory [9] of a Landau-Ginzburg model with compact critical locus $V$. We also give a precise comparison result between the current construction and Barannikov-Kontsevich's construction [2] of the Frobenius manifold structure on the cohomology of $V$.