LG–LG mirror conjecture for FJRW and Saito–Givental theories

Let $W$ be an invertible polynomial with maximal diagonal symmetry group $G_W$, and let $(W^T,\{1\})$ be its Berglund–Hübsch–Krawitz mirror pair. Write $(H_W,\bullet)$ for the FJRW state space with its product, and let $\operatorname{Jac}(W^T)$ be the Jacobian ring of $W^T$. The FJRW theory has total ancestor potential $\mathscr{A}_{W}^{\rm FJRW}$, while a choice of primitive form $\zeta$ for $W^T$ determines the Saito–Givental potential $\mathscr{A}_{W^T}^{\rm SG}$. LG–LG mirror conjecture. For a mirror pair $(W,G_W)$ and $(W^T,\{1\})$, there exists a ring isomorphism

$$(H_{W},\bullet)\cong\operatorname{Jac}(W^T)$$

together with a choice of primitive forms $\zeta$, such that the FJRW potential $\mathscr{A}_{W}^{\rm FJRW}$ is identified with the Saito–Givental potential $\mathscr{A}_{W^T}^{\rm SG}$. This is the all-genera Landau–Ginzburg mirror-symmetry prediction, identifying the A-model FJRW theory with the B-model theory arising from primitive forms and Givental’s quantization formalism; its general status is not specified in the supplied text.