Mirror Symmetry for a Cusp Polynomial Landau–Ginzburg Orbifold

Abstract

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