Mirror Symmetry for a Cusp Polynomial Landau–Ginzburg Orbifold

Abstract For any triple of positive integers $A^{\prime} = (a_1^{\prime},a_2^{\prime},a_3^{\prime})$ and $c \in{{\mathbb{C}}}^*$, cusp polynomial ${ f_{A^\prime }} x_1^{a_1^{\prime}}+x_2^{a_2^{\prime}}+x_3^{a_3^{\prime}}-c^{-1}x_1x_2x_3$ is known to be mirror Geigle–Lenzing orbifold projective line ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. More precisely, with a suitable choice primitive form, the Frobenius manifold }}$ turns out isomorphic Gromov–Witten theory In this paper we extend phenomenon equivariant case. Namely, for $G$—a symmetry group }}$, introduce pair$({ }},G)$ show that it weighted ${{\mathbb{P}}}^1_{A,\Lambda }$, indexed by another set $A$ $\Lambda $, distinct points on ${{\mathbb{C}}}\setminus \{0,1\}$. some special values $A^{\prime}$ $G$ happens ${{\mathbb{P}}}^1_{A^{\prime}} \cong{{\mathbb{P}}}^1_{A,\Lambda }$. Combining our isomorphism pair $(A,\Lambda )$, together “usual” one $A^{\prime}$, get certain identities coefficients potentials. We these are equivalent between Jacobi theta constants Dedekind eta–function.