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.

中文翻译：

---

尖点多项式 Landau-Ginzburg Orbifold 的镜像对称性

对于任意三元组正整数 $A^{\prime} = (a_1^{\prime},a_2^{\prime},a_3^{\prime})$ 和 $c \in{{\mathbb{C}} }^*$, 尖点多项式 ${ f_{A^\prime }} = x_1^{a_1^{\prime}}+x_2^{a_2^{\prime}}+x_3^{a_3^{\prime}} -c^{-1}x_1x_2x_3$ 已知是 Geigle-Lenzing 轨道双折投影线 ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^ 的镜像{\prime}}$。更确切地说，具有适当的原始形式的选择，尖端多项式$ {f_ {a ^ \ prime}} $的Frobenius歧管拒绝是Gromov-Witten理论的Frobenius歧管的同构{{\ mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$。在本文中，我们将这种镜像现象扩展到等变情况。也就是说，对于任何 $G$——一个尖点多项式 ${ f_{A^\prime }}$ 的对称群，我们引入一对 $({ f_{A^\prime }} 的 Frobenius 流形，G)$ 并证明它与 Geigle-Lenzing 加权射影线 ${{\mathbb{P}}}^1_{A,\Lambda }$ 的 Gromov-Witten 理论的 Frobenius 流形同构，由另一个集合索引$A$ 和 $\Lambda $，${{\mathbb{C}}}\setminus \{0,1\}$ 上的不同点。对于 $A^{\prime}$ 的某些特殊值以及 $G$ 的特殊选择，${{\mathbb{P}}}^1_{A^{\prime}} \cong{{\mathbb {P}}}^1_{A,\Lambda }$. 结合我们对 $(A,\Lambda )$ 的镜像对称同构，以及 $A^{\prime}$ 的“通常”同构，我们得到 Frobenius 势的系数的某些恒等式。我们证明这些恒等式等价于 Jacobi theta 常数和 Dedekind eta 函数之间的恒等式。