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 where is the Fermat quintic polynomial and . 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 .
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 , où est le polynôme de Fermat quintique et . 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 correspondant.
Accepted:
Published online:
Keywords: Landau–Ginzburg, Calabi–Yau, Mirror symmetry
Mot clés : Landau–Ginzburg, Calabi–Yau, la symétrie miroir
Priddis, Nathan 1; Shoemaker, Mark 2
@article{AIF_2016__66_3_1045_0, author = {Priddis, Nathan and Shoemaker, Mark}, title = {A {Landau{\textendash}Ginzburg/Calabi{\textendash}Yau} correspondence for the mirror quintic}, journal = {Annales de l'Institut Fourier}, pages = {1045--1091}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {3}, year = {2016}, doi = {10.5802/aif.3031}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3031/} }
TY - JOUR AU - Priddis, Nathan AU - Shoemaker, Mark TI - A Landau–Ginzburg/Calabi–Yau correspondence for the mirror quintic JO - Annales de l'Institut Fourier PY - 2016 SP - 1045 EP - 1091 VL - 66 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3031/ DO - 10.5802/aif.3031 LA - en ID - AIF_2016__66_3_1045_0 ER -
%0 Journal Article %A Priddis, Nathan %A Shoemaker, Mark %T A Landau–Ginzburg/Calabi–Yau correspondence for the mirror quintic %J Annales de l'Institut Fourier %D 2016 %P 1045-1091 %V 66 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3031/ %R 10.5802/aif.3031 %G en %F AIF_2016__66_3_1045_0
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] Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math., Volume 130 (2008) no. 5, pp. 1337-1398
[2] 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] A new cohomology theory of orbifold, Comm. Math. Phys., Volume 248 (2004) no. 1, pp. 1-31
[4] Landau-Ginzburg/Calabi–Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math. Inst. Hautes Études Sci., Volume 119 (2014), pp. 127-216
[5] Landau–Ginzburg/Calabi–Yau correspondence for quintic three–folds via symplectic transformations, Invent. Math., Volume 182 (2010) no. 1, pp. 117-165
[6] LG/CY correspondence: the state space isomorphism, Adv. Math., Volume 227 (2011) no. 6, pp. 2157-2188
[7] Twisted –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] Landau–Ginzburg/Calabi–Yau correspondence for the complete intersections and (2013) (http://arxiv.org/abs/1301.5530v3)
[9] Geometric Quantization with Applications to Gromov–Witten Theory (2013) (http://arxiv.org/abs/1309.1150)
[10] Computing genus–zero twisted Gromov–Witten invariants, Duke Math. J., Volume 147 (2009) no. 3, pp. 377-438
[11] Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, 68, American Mathematical Society, Providence, RI, 1999, xxii+469 pages
[12] The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math. (2), Volume 178 (2013) no. 1, pp. 1-106
[13] Equivariant Gromov–Witten invariants, Internat. Math. Res. Notices (1996) no. 13, pp. 613-663
[14] Symplectic geometry of Frobenius structures, Frobenius manifolds (Aspects Math., E36), Friedr. Vieweg, Wiesbaden, 2004, pp. 91-112
[15] Localization of virtual classes, Invent. Math., Volume 135 (1999) no. 2, pp. 487-518
[16] A Landau–Ginzburg mirror theorem without concavity (2013) (http://arxiv.org/abs/1307.5070, to appear in Duke Math. J.)
[17] Landau–Ginzburg/Calabi–Yau Correspondence of all Genera for Elliptic Orbifold (2011) (http://arxiv.org/abs/1106.6270)
[18] A mirror theorem for the mirror quintic, Geom. Topol., Volume 18 (2014) no. 3, pp. 1437-1483
[19] Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol., Volume 14 (2010) no. 1, pp. 1-81
[20] Catastrophes and the classification of conformal theories, Phys. Lett. B, Volume 218 (1989) no. 1, pp. 51-58
[21] Mirror manifolds and topological field theory, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 120-158
Cited by Sources: