Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $tt^*$ Structures

TL;DR

This paper rigorously proves the Calabi-Yau/Landau-Ginzburg correspondence for $tt^*$ structures, demonstrating the preservation of real structures and establishing a full correspondence between the two geometries.

Contribution

It provides a detailed analysis and modification of real structures to establish the full CY/LG correspondence for $tt^*$ geometry.

Findings

Confirmed the map preserves the $tt^*$ structures.

Modified Cecotti's real structure to ensure preservation.

Established the full CY/LG correspondence for $tt^*$ structures.

Abstract

The aim of this paper is to rigorously establish the Calabi-Yau/Landau-Ginzburg (CY/LG) correspondence for the $tt^*$ geometry structure--a generalized version of variation of Hodge structures. Although it is well-known that there exists a map between Hodge structures on the LG and CY's sides that preserves the Hodge filtration and bilinear form, it remains unclear whether the real structures are also preserved. In our paper, we conduct a detailed analysis of two period integrals on the LG's side. Based on this analysis, we modify the real structure proposed by Cecotti on LG's side, and show that the aforementioned map is also preserved under the modified real structure. As a result, we establish full CY/LG correspondence for $tt^*$ structures.