Mirror symmetry for a cusp polynomial Landau-Ginzburg orbifold

TL;DR

This paper extends mirror symmetry between cusp polynomial Landau-Ginzburg models and orbifold projective lines to an equivariant setting, establishing Frobenius manifold isomorphisms and deriving identities involving theta constants and eta functions.

Contribution

It introduces a Frobenius manifold framework for pairs of cusp polynomials and symmetry groups, generalizing mirror symmetry to equivariant cases and connecting to modular form identities.

Findings

Frobenius manifold of (f_{A'}, G) is isomorphic to Gromov-Witten theory of weighted projective lines.

Derived identities between Frobenius potential coefficients and modular functions.

Established conditions under which orbifold projective lines are isomorphic, linking to modular identities.

Abstract

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.