Year: 2021
Author: Huijun Fan, Lan Tian, Zongrui Yang
Communications in Mathematical Research , Vol. 37 (2021), Iss. 3 : pp. 297–349
Abstract
The concept of $tt^∗$ geometric structure was introduced by physicists (see [4, 10] and references therein), and then studied firstly in mathematics by C. Hertling [28]. It is believed that the $tt^∗$ geometric structure contains the whole genus 0 information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for $tt^∗$ geometry and obtain the following result. Let $f ∈ \mathbb{C}[z_0,...,z_{n+1}]$ be a nondegenerate homogeneous polynomial of degree $n$+2, then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburg model represented by a hypersurface singularity ($\mathbb{C}^{n+2}, f$), both can be written as a $tt^∗$ structure. We proved that there exists a $tt^∗$ substructure on Landau-Ginzburg side, which should correspond to the $tt^∗$ structure from variation of Hodge structures in Calabi-Yau side. We build the isomorphism of almost all structures in $tt^∗$ geometries between these two models except the isomorphism between real structures.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmr.2020-0050
Communications in Mathematical Research , Vol. 37 (2021), Iss. 3 : pp. 297–349
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 53
Keywords: $tt^∗$ geometry Landau-Ginzburg/Calabi-Yau correspondence variation of Hodge structures.