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 $c\in {\mathbb{C}}^{\ast }$, cusp polynomial $f_{A^{\prime }}=x_1^{a_1^{\prime }}+x_2^{a_2^{\prime }}+x_3^{a_3^{\prime }}-c^{-1}{x}_1{x}_2{x}_3$ is known to be mirror to Geigle-Lenzing orbifold projective line ${\mathbb{P}}_{a_1^{\prime },a_2^{\prime },a_3^{\prime }}^1$. More precisely, with a suitable choice of a primitive form, Frobenius manifold of a cusp polynomial $f_{A^{\prime }}$, turns out to be isomorphic to the Frobenius manifold of the Gromov-Witten theory of ${\mathbb{P}}_{a_1^{\prime },a_2^{\prime },a_3^{\prime }}^1$. 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 ${\mathbb{P}}_{A,\Lambda }^1$, 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 ${\mathbb{P}}_{A^{\prime }}^1\cong {\mathbb{P}}_{A,\Lambda }^1$. 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.