A Landau–Ginzburg/Calabi–Yau correspondence for the mirror quintic
Annales de l'Institut Fourier, Volume 66 (2016) no. 3, pp. 1045-1091.

We prove a version of the Landau–Ginzburg/Calabi–Yau correspondence for the mirror quintic. In particular we calculate the genus–zero FJRW theory for the pair (W,G) where W is the Fermat quintic polynomial and G=SL W . We identify it with the Gromov–Witten theory of the mirror quintic three–fold via an explicit analytic continuation and symplectic transformation. In the process we prove a mirror theorem for the corresponding Landau–Ginzburg model (W,G).

Nous montrons une version de la correspondance Landau–Ginzburg/ Calabi–Yau pour le miroir quintique. Plus précisément, on calcule la théorie FJRW en genre zéro pour la paire (W,G), où W est le polynôme de Fermat quintique et G=SL W . On l’identifie ensuite avec la théorie de Gromov–Witten de la quintique avec une continuation analytique explicite et une transformation symplectique. On démontre au passage un théorème miroir pour le modèle de Landau–Ginzburg (W,G) correspondant.

Published online:
DOI: 10.5802/aif.3031
Classification: 14N35, 14J33, 53D45, 14J17, 32G20
Keywords: Landau–Ginzburg, Calabi–Yau, Mirror symmetry
Mot clés : Landau–Ginzburg, Calabi–Yau, la symétrie miroir
Priddis, Nathan 1; Shoemaker, Mark 2

1 Mathematics Department University of Michigan 530 Church St Ann Arbor, MI 48109 (USA)
2 Department of Mathematics University of Utah Salt Lake City, UT 84112 (USA)
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Priddis, Nathan; Shoemaker, Mark. A Landau–Ginzburg/Calabi–Yau correspondence for the mirror quintic. Annales de l'Institut Fourier, Volume 66 (2016) no. 3, pp. 1045-1091. doi : 10.5802/aif.3031. https://aif.centre-mersenne.org/articles/10.5802/aif.3031/

[1] Abramovich, Dan; Graber, Tom; Vistoli, Angelo Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math., Volume 130 (2008) no. 5, pp. 1337-1398

[2] 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

[3] Chen, Weimin; Ruan, Yongbin A new cohomology theory of orbifold, Comm. Math. Phys., Volume 248 (2004) no. 1, pp. 1-31

[4] 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), pp. 127-216

[5] 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

[6] Chiodo, Alessandro; Ruan, Yongbin LG/CY correspondence: the state space isomorphism, Adv. Math., Volume 227 (2011) no. 6, pp. 2157-2188

[7] Chiodo, Alexander; Zvonkine, Dimitri Twisted r–spin potential and Givental’s quantization, Adv. Theor. Math. Phys., Volume 13 (2009) no. 5, pp. 1335-1369 http://projecteuclid.org.proxy.lib.umich.edu/getRecord?id=euclid.atmp/1282054097

[8] Clader, Emily Landau–Ginzburg/Calabi–Yau correspondence for the complete intersections X 3,3 and X 2,2,2,2 (2013) (http://arxiv.org/abs/1301.5530v3)

[9] Clader, Emily; Priddis, Nathan; Shoemaker, Mark Geometric Quantization with Applications to Gromov–Witten Theory (2013) (http://arxiv.org/abs/1309.1150)

[10] Coates, Tom; Corti, Alessio; Iritani, Hiroshi; Tseng, Hsian–Hua Computing genus–zero twisted Gromov–Witten invariants, Duke Math. J., Volume 147 (2009) no. 3, pp. 377-438

[11] Cox, David A.; Katz, Sheldon Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, 68, American Mathematical Society, Providence, RI, 1999, xxii+469 pages

[12] 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

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

[14] Givental, Alexander B. Symplectic geometry of Frobenius structures, Frobenius manifolds (Aspects Math., E36), Friedr. Vieweg, Wiesbaden, 2004, pp. 91-112

[15] Graber, T.; Pandharipande, R. Localization of virtual classes, Invent. Math., Volume 135 (1999) no. 2, pp. 487-518

[16] Guéré, Jérémy A Landau–Ginzburg mirror theorem without concavity (2013) (http://arxiv.org/abs/1307.5070, to appear in Duke Math. J.)

[17] Krawitz, Marc; Shen, Yefeng Landau–Ginzburg/Calabi–Yau Correspondence of all Genera for Elliptic Orbifold 1 (2011) (http://arxiv.org/abs/1106.6270)

[18] Lee, Yuan-Pin; Shoemaker, Mark A mirror theorem for the mirror quintic, Geom. Topol., Volume 18 (2014) no. 3, pp. 1437-1483

[19] Tseng, Hsian–Hua Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol., Volume 14 (2010) no. 1, pp. 1-81

[20] Vafa, Cumrun; Warner, Nicholas Catastrophes and the classification of conformal theories, Phys. Lett. B, Volume 218 (1989) no. 1, pp. 51-58

[21] Witten, Edward Mirror manifolds and topological field theory, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 120-158

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