Global mirror symmetry for invertible simple elliptic singularities

Milanov, Todor; Shen, Yefeng

Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 271-330.

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.

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.

Classification : 14N35, 14B05
Keywords: mirror symmetry, simple elliptic singularities
Mot clés : symétrie miroir, singularités simples elliptiques

[1] Acosta, Pedro FJRW-Rings and Landau-Ginzburg mirror symmetry in two dimensions (http://arxiv.org/abs/0906.0970)

[2] Arnolʼd, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. II, Monographs in Mathematics, 83, Birkhäuser Boston, Inc., Boston, MA, 1988, viii+492 pages (Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi) | DOI

[3] Berglund, Per; Hübsch, Tristan A generalized construction of mirror manifolds, Nuclear Phys. B, Volume 393 (1993) no. 1-2, pp. 377-391 | DOI

[4] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C. Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys., Volume 165 (1994) no. 2, pp. 311-427 http://projecteuclid.org/euclid.cmp/1104271134

[5] Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B, Volume 359 (1991) no. 1, pp. 21-74 | DOI

[6] Chen, Weimin; Ruan, Yongbin Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001) (Contemp. Math.), Volume 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25-85 | DOI

[7] Chiodo, Alessandro Towards an enumerative geometry of the moduli space of twisted curves and $r$th roots, Compos. Math., Volume 144 (2008) no. 6, pp. 1461-1496 | DOI

[8] Chiodo, Alessandro; Iritani, Hiroshi; Ruan, Yongbin Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math. Inst. Hautes Études Sci., Volume 119 (2014) no. 1, pp. 127-216 | DOI

[9] Chiodo, Alessandro; Ruan, Yongbin Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations, Invent. Math., Volume 182 (2010) no. 1, pp. 117-165 | DOI

[10] Chiodo, Alessandro; Ruan, Yongbin A global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yau correspondence, Ann. Inst. Fourier (Grenoble), Volume 61 (2011) no. 7, pp. 2803-2864 http://aif.cedram.org/item?id=AIF_2011__61_7_2803_0

[11] Coates, Tom; Iritani, Hiroshi On the Convergence of Gromov-Witten Potentials and Givental’s Formula (http://arxiv.org/abs/1203.4193v1)

[12] Costello, Kevin; Li, Si Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model (http://arxiv.org/abs/1201.4501v1)

[13] Doran, Charles F.; Morgan, John W. Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi-Yau threefolds, Mirror symmetry. V (AMS/IP Stud. Adv. Math.), Volume 38, Amer. Math. Soc., Providence, RI, 2006, pp. 517-537

[14] Dubrovin, Boris Geometry of $2$D topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993) (Lecture Notes in Math.), Volume 1620, Springer, Berlin, 1996, pp. 120-348 | DOI

[15] Fan, Huijun; Jarvis, Tyler; Ruan, Yongbin The Witten Equation and Its Virtual Fundamental Cycle (book in preparation, http://arxiv.org/abs/0712.4025v3)

[16] Fan, Huijun; Jarvis, Tyler; Ruan, Yongbin The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math. (2), Volume 178 (2013) no. 1, pp. 1-106 | DOI

[17] Fan, Huijun; Shen, Yefeng Quantum ring of singularity $X^p+X{Y}^q$, Michigan Math. J., Volume 62 (2013) no. 1, pp. 185-207 | DOI

[18] Gährs, Swantje Picard-Fuchs equations of special one-parameter families of invertible polynomials, Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds (Fields Inst. Commun.), Volume 67, Springer, New York, 2013, pp. 285-310 | DOI

[19] Givental, Alexander B. Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices (1996) no. 13, pp. 613-663 | DOI

[20] Givental, Alexander B. Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J., Volume 1 (2001) no. 4, p. 551-568, 645 (Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary)

[21] Givental, Alexander B. Semisimple Frobenius structures at higher genus, Internat. Math. Res. Notices (2001) no. 23, pp. 1265-1286 | DOI

[22] Hertling, Claus Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, 151, Cambridge University Press, Cambridge, 2002, x+270 pages | DOI

[23] Huang, M.-x.; Klemm, A.; Quackenbush, S. Topological string theory on compact Calabi-Yau: modularity and boundary conditions, Homological mirror symmetry (Lecture Notes in Phys.), Volume 757, Springer, Berlin, 2009, pp. 45-102

[24] Krawitz, Marc FJRW rings and Landau-Ginzburg Mirror Symmetry (http://arxiv.org/abs/0906.0796v1)

[25] Krawitz, Marc; Shen, Yefeng Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold ${\mathbb{P}}^1$ (http://arxiv.org/abs/1106.6270)

[26] Li, Si On the quantum theory of Landau-Ginzburg B-model (preprint)

[27] Lian, Bong H.; Liu, Kefeng; Yau, Shing-Tung Mirror principle. I, Asian J. Math., Volume 1 (1997) no. 4, pp. 729-763

[28] Manin, Yuri I. Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, 47, American Mathematical Society, Providence, RI, 1999, xiv+303 pages

[29] Milanov, Todor Analyticity of the total ancestor potential in singularity theory, Adv. Math., Volume 255 (2014), pp. 217-241 | DOI

[30] Milanov, Todor; Ruan, Yongbin Gromov-Witten theory of elliptic orbifold ${\mathbb{P}}^1$ and quasi-modular forms (http://arxiv.org/abs/1106.2321v1)

[31] Milanov, Todor; Ruan, Yongbin; Shen, Yefeng Gromov–Witten theory and cycled-valued modular forms http://arxiv.org/abs/1206.3879v1, to appear in Journal für die reine und angewandte Mathematik (Crelles Journal). DOI: 10.1515/crelle-2015-0019

[32] Milanov, Todor; Shen, Yefeng The modular group for the total ancestor potential of Fermat simple elliptic singularities, Commun. Number Theory Phys., Volume 8 (2014) no. 2, pp. 329-368 | DOI

[33] Ruan, Yongbin The Witten equation and the geometry of the Landau-Ginzburg model, String-Math 2011 (Proc. Sympos. Pure Math.), Volume 85, Amer. Math. Soc., Providence, RI, 2012, pp. 209-240 | DOI

[34] Saito, Kyoji On Periods of Primitive Integrals, I Preprint RIMS(1982)

[35] Saito, Kyoji Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Volume 14 (1971), pp. 123-142

[36] Saito, Kyoji Einfach-elliptische Singularitäten, Invent. Math., Volume 23 (1974), pp. 289-325

[37] Saito, Kyoji Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 28 (1981) no. 3, p. 775-792 (1982)

[38] Saito, Kyoji; Takahashi, Atsushi From primitive forms to Frobenius manifolds, From Hodge theory to integrability and TQFT tt*-geometry (Proc. Sympos. Pure Math.), Volume 78, Amer. Math. Soc., Providence, RI, 2008, pp. 31-48 | DOI

[39] Satake, Ikuo; Takahashi, Atsushi Gromov-Witten invariants for mirror orbifolds of simple elliptic singularities, Ann. Inst. Fourier (Grenoble), Volume 61 (2011) no. 7, pp. 2885-2907 http://aif.cedram.org/item?id=AIF_2011__61_7_2885_0

[40] Teleman, Constantin The structure of 2D semi-simple field theories, Invent. Math., Volume 188 (2012) no. 3, pp. 525-588 | DOI