数理解析研究所讲究録第1918巻.PDFVIP

数理解析研究所講究録 第 1918 巻 2014 年 59-70 59 FROM PRIMITIVE FORM TO MIRROR SYMMETRY KYOJI SAITO ABSTRACT. This is a report on the recent joint work [29] on LG-LG mirror symmetry for the 14 exceptional unimodular singularities.1 1. INTR0DUCT10N 1 Present note is worked out with the help of the coauthors Changzheng Li, Si Li and Yefeng Shen, to whom the author expresses his deep gratitudes. 60 Primitive forms are about universal deformations $F$ of functions, giving ?at structures on the deformation spaces. Hence, the theory is relevant in the complex geometric (B- model) aspects of $N=(2,2)$ supersymmetric Landau-Ginzburg (LG) theory with the su- perpotential $F$ . However, this pattern of the mirror symmetry was not mathematically rigorously worked out until recently. This is because of (1) lack of mathematical theory of $A$-model LG-theory at that early time, and (2) the di?culty of calculating primitive forms until recently, where explicit expressions of primitive forms were known only for weighted homogeneous polynomials of central charge less than or equal to 1. Both di?- culties were resolved as follows. In around 2007, Fan, Jarvis and Ruan constructed a so-called quantum singularity theory by counting virtual cycles associated with a weighted homogeneous polynomial, whose potential (generating series) gives again a Frobenius manifold structure on the so-called $FJR\mathcal{W}$ state space [14, 15]. This is considered as an $A$-model Landau-Ginzburg theory. They immediately realized that such Frobenius manifold for an ADE-polynomial $W$ is actually isomorphic to the Frobenius manifold arising from the primitive form (i.e. B- side) of another ADE-polynomial $\mathcal{W}^{T}$ . The superpotent