On the Geometry of Landau-Ginzburg Model

Speaker 

Huijun Fan got PhD in Peking University in 1998, and then did postdoctoral works in Max-Planck Institute in Leipzig and in Mathematics Institute of AMSS. He joined Peking University in 2003 and became a full professor in 2010. He was the dean of Math. department in 2017-2021. Currently he is the director of the Key Laboratory of Mathematics and Applied Mathematics of the Ministry of Education of Peking University and the deputy director of the Sino-Russian Math Center. He has won National Outstanding Youth Grant, Changjiang Distinguished Professor of the Ministry of Education, and the Second Prize of the National Natural Science Award. He is the plenary speaker of the 2021 annual meeting of the Chinese Mathematical Society.

Abstract

An LG model (M, f) is given by a noncompact complex manifold M and the holomorphic function f defined on it, which is an important model in string theory. Because of the mirror symmetry conjecture, the research on the geometric structure and quantization theory of LG model has attracted more and more attention. Given a Calabi-Yau (CY) manifold, we can define Gromov-Witten theory (A theory) on it, and also study the variation of Hodge structure on its mirror manifold (B theory). Accordingly, LG model includes A theory - FJRW theory and Hodge structure variational theory.

This report starts with some examples, gives the geometric and topological information contained by a LG model, and derives the relevant Witten equation (nonlinear) and Schrodinger equation (linear). The study of the solution space of these two sets of equations will lead to different quantization theories. Secondly, we give our recent correspondence theorem of Hodge structures between LG model and CY manifold. Finally, we will discuss some relevant issues.