Mirror Symmetry for a Cusp Polynomial Landau-Ginzburg Orbifold

For any triple of positive integers A' = (a'(1), a'(2), a'(3)) and c is an element of C*, cusp polynomial f(A)' = x(1)(a'1) + x(2)(a'2) + x(3)(a'3) - c(-1) x(1)x(2)x(3) is known to be mirror to Geigle-Lenzing orbifold projective line P-a'1,a'2,a'3(1). More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial f(A') turns out to be isomorphic to the Frobenius manifold of the Gromov-Witten theory of P-a'1,a'2,a'3(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'), 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 P-A,Lambda(1), indexed by another set A and Lambda, distinct points on C \ {0, 1}. For some special values of A' with the special choice of G it happens that P-A'(1) congruent to P-A,Lambda(1). 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.