Categorical Saito theory, II: Landau-Ginzburg orbifolds

Let $W\in \mathbb{C}[x_1,\cdots ,x_N]$ be an invertible polynomial with an isolated singularity at origin, and let $G\subset {{\sf SL}}_N\cap ($\mathbb{C}$^*)^N$beafinitediagonalandspeciallinearsymmetrygroupof$W$\mathrm{.}Inthispaper,weusethecategory${{\sf MF}}_G(W)$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 ${\mathsf{M}\mathsf{F}}_G(W)$ characterized by $G_W^{{\sf max}}$-equivariance\mathrm{.}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 $(\frac{1}{5}(x_1^5+\cdots +x_5^5),\mathbb{Z}\mathrm{/}5\mathbb{Z})$ with its mirror dual B-model Landau-Ginzburg orbifold $(\frac{1}{5}(x_1^5+\cdots +x_5^5),(\mathbb{Z}\mathrm{/}5\mathbb{Z}{)}^4)$. In the case of the Quintic family $$\mathcal{W}$=\frac{1}{5}(x_1^5+\cdots+x_5^5)-\psi x_1x_2x_3x_4x_5$, we also prove a comparison result of B-model VSHS’s conjectured by Ganatra-Perutz-Sheridan.