A Landau-Ginzburg/Calabi-Yau Correspondence for the Mirror Quintic.

Abstract

From High Energy Physics there is a certain expected correspondence between two different physical models, the Landau--Ginzburg model and the geometric or Calabi--Yau model. This correspondence is known as the Landau--Ginzburg/Calabi--Yau correspondence. The Landau--Ginzburg model has been recently developed mathematically, and goes by the name of FJRW theory. The Calabi--Yau model has been around much longer and is known mathematically as Gromov--Witten theory. The Landau--Ginzburg/Calabi--Yau correspondence therefore takes the form of a precise relationship between the Gromov--Witten theory of a Calabi--Yau variety and the FJRW theory for a certain related potential function. We state and prove the relationship for the famous "mirror quintic"---a quotient of the quintic three--fold by a certain finite group. In the process we also prove a Landau--Ginzburg mirror theorem for the mirror quintic.