CY/LG correspondence for Weil-Petersson metrics and tt⁎ structures

Abstract

The purpose of this paper is to establish the Calabi-Yau/Landau-Ginzburg (CY/LG) correspondence for the $t{t}^{⁎}$ geometry structure, which is a generalized version of the variation of Hodge structures. To begin, consider a homogeneous polynomial $f:C^n\to C$ of degree n. We can put a natural Hodge structure on the space of harmonic forms with respect to the twisted Laplacian ${\Delta }_f$ (cf. §3.1). Additionally, there exists a natural Hodge structure on the cohomology of the Calabi-Yau hypersurface defined by f in the projective space $P^{n-1}$. Naturally, one may question the relationship between these two Hodge structures and, more generally, the connection between the corresponding $t{t}^{⁎}$ structures. As an application of our main result, we modify the real structure proposed by Cecotti on the Jacobi ring of f (cf. [8, (4.2)]). We show that this modified real structure not only aligns with pole-order filtration and Grothendieck residue pairing on the Jacobi ring of f, but it is also preserved by the residue map constructed by Griffiths-Carlson. Finally, it is crucial to emphasize that the CY/LG correspondence for the $t{t}^{⁎}$ structures establishes the fundamental basis for studying the CY/LG correspondence for the genus one terms in the B-model.