Gromov--Witten invariants for mirror orbifolds of simple elliptic singularities

Abstract

We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov--Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.

20 pages; some explanations and references added, the proof of Lemma 3.4 clarified, typos added. This paper will appear at Annales de l'Institut Fourier in Volume special 61.7

References in corpus (5)

• Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms
• Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold $p^1$
• Homological Mirror Symmetry and Simple Elliptic Singularities
• Simple Elliptic Singularities: a note on their G-function
• Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations

Cited by in corpus (9)

• Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms
• Primitive Forms for Affine Cusp Polynomials
• Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}
• Simple Elliptic Singularities: a note on their G-function
• Polynomial Modular Frobenius Manifolds
• On quantum cohomology ring of elliptic ${\mathbb{P}}^1$ orbifolds
• Counting of Holomorphic Orbi-spheres in ${\mathbb{P}}_{2,2,2,2}^1$ and Determinant Equation
• On Frobenius Manifolds from Gromov--Witten Theory of Orbifold Projective Lines with $r$ orbifold points
• A Uniqueness Theorem for Frobenius Manifolds and Gromov--Witten Theory for Orbifold Projective Lines