Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities

Satake, Ikuo; Takahashi, Atsushi

Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2885-2907.

Nous considérons une symétrie miroir des singularités elliptiques simples. En particulier, nous construisons des isomorphismes de variétés de Frobenius entre celui de la théorie de Gromov–Witten d’une droite projective à poids, celui de la théorie des formes primitives pour un déploiement universel d’une singularité elliptique simple et celui de la théorie des invariants pour un groupe de Weyl elliptique. Comme conséquence, nous donnons une interprétation géométrique des coefficients de Fourier d’un produit eta considéré par K. Saito.

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.

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