Global mirror symmetry for invertible simple elliptic singularities
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, p. 271-330

A simple elliptic singularity can be described in terms of a marginal deformation of an invertible polynomial. The choice of the polynomials and its marginal deformation are not unique. In this paper, following the earlier work of Krawitz-Shen and Milanov-Ruan, we investigate the global mirror symmetry phenomenon for simple elliptic singularities. We prove that the mirror symmetry for each family is governed by a certain system of hypergeometric equations. We conjecture that the Saito-Givental theory of the family at any special limit is mirror to either the Gromov-Witten theory of an elliptic orbifold projective line or the Fan-Jarvis-Ruan-Witten theory of an invertible polynomial, and the limits are classified by the Milnor number of the singularity and the j-invariant at the special limit. We prove the conjecture holds at all special limits of the Fermat polynomials and at the Gepner points in all other cases.

Une singularité simple elliptique peut être décrite en termes d’une déformation marginale d’un polynôme inversible. Le choix du polynôme et de sa déformation n’est pas unique. Dans ce papier, suivant les travaux de Krawitz-Shen et Milanov-Ruan, nous regardons la symétrie miroir globale pour les singularités simples elliptiques. Nous prouvons que la symétrie miroir pour chaque famille est règlée par un certain système d’équations hypergéométriques. Nous conjecturons que la théorie de Saito-Givental de la famille à une limite spéciale est liée soit à la théorie de Gromov-Witten d’une droite projective orbifold elliptique, soit à la théorie Fan-Jarvis-Ruan-Witten d’un polynôme inversible. Les limites sont classifiées par le nombre de Milnor de la singularité, et par le j-invariant à la limite spéciale. Nous vérifions la conjecture pour toutes les limites spéciales des polynômes de Fermat, et pour tous les points de Gepner dans les autres cas.

Received : 2014-05-19
Revised : 2015-02-07
Accepted : 2015-05-07
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3012
Classification:  14N35,  14B05
Keywords: mirror symmetry, simple elliptic singularities
@article{AIF_2016__66_1_271_0,
     author = {Milanov, Todor and Shen, Yefeng},
     title = {Global mirror symmetry for invertible simple elliptic singularities},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {271-330},
     doi = {10.5802/aif.3012},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_1_271_0}
}
Milanov, Todor; Shen, Yefeng. Global mirror symmetry for invertible simple elliptic singularities. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 271-330. doi : 10.5802/aif.3012. http://www.numdam.org/item/AIF_2016__66_1_271_0/

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