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

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 $W$.