Primitive forms via polyvector fields

TL;DR

This paper introduces a complex geometric framework for primitive forms and higher residues, providing explicit constructions and algorithms that advance understanding in mirror symmetry and singularity theory.

Contribution

It develops a new geometric approach to primitive forms, offering explicit perturbative methods and a comprehensive description of their moduli space for weighted homogeneous cases.

Findings

Explicit perturbative expressions for primitive forms of E_12 singularity

Algorithm for computing Taylor expansions of primitive forms

Application to Landau-Ginzburg mirror symmetry with FJRW-theory

Abstract

We develop a complex differential geometric approach to the theory of higher residues and primitive forms from the viewpoint of Kodaira-Spencer gauge theory, unifying the semi-infinite period maps for Calabi-Yau models and Landau-Ginzburg models. We give an explicit perturbative construction of primitive forms with respect to opposite filtrations and primitive elements. This leads to a concrete algorithm to compute the Taylor expansions of primitive forms as well as the description of their moduli space for all weighted homogenous cases. As an example, we present unknown perturbative expressions for the primitive form of E_12 singularity and illustrate its application to Landau-Ginzburg mirror symmetry with FJRW-theory.