Mirror Symmetry for a Cusp Polynomial Landau–Ginzburg Orbifold

International Mathematics Research Notices. 2022. Vol. 2022. No. 19. P. 14865–14922.

Basalaev A., Takahashi A.

For any triple of positive integers A′=(a′1,a′2,a′3) and c∈C∗⁠, cusp polynomial fA′=xa′11+xa′22+xa′33−c−1x1x2x3 is known to be mirror to Geigle–Lenzing orbifold projective line P1a′1,a′2,a′3⁠. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial fA′ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of P1a′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 fA′⁠, we introduce the Frobenius manifold of a pair  (fA′,G) and show that it is isomorphic to the Frobenius manifold of the Gromov–Witten theory of Geigle–Lenzing weighted projective line P1A,Λ⁠, indexed by another set A and Λ⁠, distinct points on C∖{0,1}⁠. For some special values of A′ with the special choice of G it happens that P1A′≅P1A,Λ⁠. Combining our mirror symmetry isomorphism for the pair (A,Λ)⁠, 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.

Keywords: singularity theory, Frobenius manifolds, Mirror symmetry