Categorical primitive forms and Gromov-Witten invariants of $A_n$ singularities

TL;DR

This paper introduces a categorical analogue of primitive forms for $A_n$ singularities and proves the uniqueness of such forms in the matrix factorizations category, linking categorical Gromov-Witten invariants with classical singularity invariants.

Contribution

It defines a categorical primitive form for $A_n$ singularities and establishes its uniqueness, connecting categorical invariants with classical singularity invariants.

Findings

Existence and uniqueness of categorical primitive forms for $A_n$ singularities.

Categorical Gromov-Witten invariants match classical invariants from singularity unfolding.

Abstract

We introduce a categorical analogue of Saito's notion of primitive forms. Let $W$ denote the potential $\frac{1}{n+1} x^{n+1}$. For the category $MF(W)$ of matrix factorizations of $W$ we prove that there exists a unique, up to non-zero constant, categorical primitive form. The corresponding genus zero categorical Gromov-Witten invariants of $MF(W)$ are shown to match with the invariants defined through unfolding of singularities of $W$.