Categorical Saito theory, II: Landau-Ginzburg orbifolds

Abstract

Let $W\in \mathbb{C}[x_1,\cdots ,x_N]$ be an invertible polynomial with an isolated singularity at origin, and let $G\subset {\mathsf{SL}}_N\cap ({\mathbb{C}}^{\ast }{)}^N$ be a finite diagonal and special linear symmetry group of $W$. In this paper, we use the category ${\mathsf{MF}}_G(W)$ of $G$-equivariant matrix factorizations and its associated VSHS to construct a $G$-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of ${\mathsf{MF}}_G(W)$ characterized by $G_W^{\mathsf{max}}$-equivariance. Conjecturally, this $G$-equivariant Saito theory is equivalent to the genus zero part of the FJRW theory under LG/LG mirror symmetry. In the marginal deformation direction, we verify this for the FJRW theory of $(\frac{1}{5}(x_1^5+\cdots +x_5^5),\mathbb{Z}/5\mathbb{Z})$ with its mirror dual B-model Landau-Ginzburg orbifold $(\frac{1}{5}(x_1^5+\cdots +x_5^5),(\mathbb{Z}/5\mathbb{Z}{)}^4)$. In the case of the Quintic family $\mathcal{W}=\frac{1}{5}(x_1^5+\cdots +x_5^5)-\psi x_1{x}_2{x}_3{x}_4{x}_5$, we also prove a comparison result of B-model VSHS's conjectured by Ganatra-Perutz-Sheridan.

30 pages, comments welcome! Minor modification in V2