For any triple of positive integers $A' = (a_1',a_2',a_3')$ and $c \in \mathbb{ C}^*$, cusp polynomial $f_{ A'} = x_1^{ a_1'}+x_2^{ a_2'}+x_3^{ a_3'}-c^{ -1}x_1x_2x_3$ is known to be mirror to Geigle-Lenzing orbifold projective line $\mathbb{ P}^1_{ a_1',a_2',a_3'}$. More precisely, with a suitable choice of a primitive form, Frobenius manifold of a cusp polynomial $f_{ A'}$, turns out to be isomorphic to the Frobenius manifold of the Gromov-Witten theory of $\mathbb{ P}^1_{ a_1',a_2',a_3'}$. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any $G$ - a symmetry group of a cusp polynomial $f_{ A'}$, we introduce the Frobenius manifold of a pair $(f_{ A'},G)$ 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 $A'$ with the special choice of $G$ it happens that $\mathbb{ P}^1_{ A'} \cong \mathbb{ P}^1_{ A,\Lambda}$. Combining our mirror symmetry isomorphism for the pair $(A,\Lambda)$, together with the "usual" one for $A'$, 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.
Categories: math.AG math-ph math.MP