Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities
Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2885-2907.

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.

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.

DOI: 10.5802/aif.2797
Classification: 14J33, 14N35, 32S25
Keywords: a mirror symmetry, simple elliptic singularities, Frobenius manifolds, Gromov–Witten theory, weighted projective line, primitive forms, the invariant theory, an elliptic Weyl group, an eta product
Mot clés : symétrie miroir, singularité elliptique simple, variété de Frobenius, théorie de Gromov–Witten, droite projective à poids, forme primitive, théorie des invariants, groupe de Weyl elliptique, produit eta
Satake, Ikuo 1; Takahashi, Atsushi 2

1 Faculty of Education, Kagawa University, 1-1 Saiwai-cho Takamatsu Kagawa, 760-8522, Japan
2 Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka Osaka, 560-0043, Japan
     author = {Satake, Ikuo and Takahashi, Atsushi},
     title = {Gromov{\textendash}Witten invariants for mirror orbifolds of simple elliptic singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {2885--2907},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {7},
     year = {2011},
     doi = {10.5802/aif.2797},
     zbl = {1294.14016},
     mrnumber = {3112511},
     language = {en},
     url = {}
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Satake, Ikuo; Takahashi, Atsushi. Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities. Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2885-2907. doi : 10.5802/aif.2797.

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