@Article{CMR-37-3,
author = {Huijun, Fan and Tian, Lan and Zongrui, Yang},
title = {LG/CY Correspondence Between $tt^∗$ Geometries},
journal = {Communications in Mathematical Research },
year = {2021},
volume = {37},
number = {3},
pages = {297--349},
abstract = {<p style="text-align: justify;">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 &nbsp;$tt^∗$ 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.</p>},
issn = {2707-8523},
doi = {https://doi.org/10.4208/cmr.2020-0050},
url = {https://global-sci.com/article/81510/lgcy-correspondence-between-tt-geometries}
}