TY - JOUR
T1 - LG/CY Correspondence Between $tt^∗$ Geometries
AU - Fan , Huijun
AU - Tian , Lan
AU - Yang , Zongrui
JO - Communications in Mathematical Research 
VL - 3
SP - 297
EP - 349
PY - 2021
DA - 2021/06
SN - 37
DO - http://doi.org/10.4208/cmr.2020-0050
UR - https://global-sci.org/intro/article_detail/cmr/19264.html
KW - $tt^∗$ geometry, Landau-Ginzburg/Calabi-Yau correspondence, variation of
Hodge structures.
AB - <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>