A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence
[Un cadre de symétrie miroir globale pour la correspondance Landau–Ginzburg/Calabi–Yau]
Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2803-2864.

On montre comment la correspondance Landau–Ginzburg/Calabi–Yau pour la variété quintique dans 4 s’inscrit naturellement dans un cadre de symétrie miroir globale. On s’inspire de la dualité miroir de Berglund–Hübsch pour fournir un cadre conjectural analogue qui incorpore toutes les hypersurfaces de Calabi–Yau dans les espaces projectifs à poids, ainsi que certains quotients par l’action de groupes abéliens finis.

We show how the Landau–Ginzburg/Calabi–Yau correspondence for the quintic three-fold can be cast into a global mirror symmetry framework. Then we draw inspiration from Berglund–Hübsch mirror duality construction to provide an analogue conjectural picture featuring all Calabi–Yau hypersurfaces within weighted projective spaces and certain quotients by finite abelian group actions.

DOI : 10.5802/aif.2795
Classification : 14J33, 14J32, 14H10
Keywords: Mirror symmetry, Gromov–Witten theory, Calabi–Yau varieties, moduli of curves
Mot clés : Symétrie miroir, théorie de Gromov–Witten, variétés de Calabi–Yau, modules de courbes
Chiodo, Alessandro 1 ; Ruan, Yongbin 2

1 Institut de Mathématiques de Jussieu UMR 7586 CNRS Université Pierre et Marie Curie Case 247 4 Place Jussieu 75252 Paris cedex 05 France
2 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA
@article{AIF_2011__61_7_2803_0,
     author = {Chiodo, Alessandro and Ruan, Yongbin},
     title = {A global mirror symmetry framework for the {Landau{\textendash}Ginzburg/Calabi{\textendash}Yau} correspondence},
     journal = {Annales de l'Institut Fourier},
     pages = {2803--2864},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {7},
     year = {2011},
     doi = {10.5802/aif.2795},
     mrnumber = {3112509},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2795/}
}
TY  - JOUR
AU  - Chiodo, Alessandro
AU  - Ruan, Yongbin
TI  - A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 2803
EP  - 2864
VL  - 61
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2795/
DO  - 10.5802/aif.2795
LA  - en
ID  - AIF_2011__61_7_2803_0
ER  - 
%0 Journal Article
%A Chiodo, Alessandro
%A Ruan, Yongbin
%T A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence
%J Annales de l'Institut Fourier
%D 2011
%P 2803-2864
%V 61
%N 7
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2795/
%R 10.5802/aif.2795
%G en
%F AIF_2011__61_7_2803_0
Chiodo, Alessandro; Ruan, Yongbin. A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence. Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2803-2864. doi : 10.5802/aif.2795. http://www.numdam.org/articles/10.5802/aif.2795/

[1] Abramovich, D.; Graber, T.; Vistoli, A. Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math., Volume 130 (2008) no. 5 | MR | Zbl

[2] Abramovich, D.; Jarvis, T. J. Moduli of twisted spin curves, Proc. Amer. Math. Soc., Volume 131 (2003), pp. 685-699 (preprint version: math.AG/0104154) | MR | Zbl

[3] Aganagic, M.; Bouchard, V.; Klemm, A. Topological Strings and (Almost) Modular Forms, Commun. Math. Phys., Volume 277 (2008), pp. 771-819 (preprint version: hep-th/0607100) | MR | Zbl

[4] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps, II, Birkhäuser, Boston, 1988 | MR

[5] Batyrev, V. V.; Borisov, L. A. Dual Cones and Mirror Symmetry for Generalized Calabi–Yau Manifolds, Mirror Symmetry II (AMS/IP Stud. Adv. Math 1), Amer. Math. Soc. Providence, RI, 1997, pp. 71-86 | MR | Zbl

[6] Berglund, P.; Hübsch, T. A Generalized Construction of Mirror Manifolds, Nuclear Physics B, Volume 393 (1993), p. 397-391 | MR | Zbl

[7] Berglund, P.; Katz, S. Mirror Symmetry Constructions: A Review, Mirror Symmetry II (AMS/IP Stud. Adv. Math 1), Amer. Math. Soc. Providence, RI, 1997, pp. 87-113 (preprint version: arXiv:hep-th/9406008) | MR | Zbl

[8] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C. Kodaira–Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Comm. Math. Phy., Volume 165 (1994), pp. 311-427 | MR | Zbl

[9] Boissière, S.; Mann, É; Perroni, F. A model for the orbifold Chow ring of weighted projective spaces, Communications in Algebra, Volume 37 (2009), pp. 503-514 | MR | Zbl

[10] Borisov, L. Berglund–Hübsch mirror symmetry via vertex algebras (preprint version: arXiv:1007.2633v3)

[11] Candelas, P.; De La Ossa, X. C.; Green, P. S.; Parkes, L. A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B, Volume 359 (1991), pp. 21-74 | MR | Zbl

[12] Chiodo, A. The Witten top Chern class via K-theory, J. Algebraic Geom., Volume 15 (2006) no. 4, pp. 681-707 (preprint version: math.AG/0210398) | MR | Zbl

[13] Chiodo, A. Stable twisted curves and their r-spin structures (Courbes champêtres stables et leurs structures r-spin), Ann. Inst. Fourier, Volume 58 (2008) no. 5, pp. 1635-1689 (preprint version: math.AG/0603687) | Numdam | MR | Zbl

[14] Chiodo, A. Towards an enumerative geometry of the moduli space of twisted curves and rth roots, Compos. Math., Volume 144 (2008), pp. 1461-1496 (Part 6, preprint version: math.AG/0607324) | MR | Zbl

[15] Chiodo, A.; Iritani, H.; Ruan, Y. Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence (preprint version: arXiv:1201.0813)

[16] Chiodo, A.; Ruan, Y. Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations, Invent. Math., Volume 182 (2010), pp. 117-165 (preprint version: arXiv:0812.4660) | MR | Zbl

[17] Chiodo, A.; Ruan, Y. LG/CY correspondence: the state space isomorphism, Adv. Math., Volume 227 (2011) no. 6, pp. 2157-2188 | MR | Zbl

[18] Chiodo, A.; Zvonkine, D. Twisted Gromov–Witten r-spin potentials and Givental’s quantization, Adv. Theor. Math. Phys., Volume 13 (2009) no. 5, pp. 1335-1369 (preprint version: arXiv:0711.0339) | MR | Zbl

[19] Clarke, P. Duality for toric Landau-Ginzburg models (preprint: arXiv:0803.0447)

[20] Coates, T. On the Crepant Resolution Conjecture in the Local Case, Communications in Mathematical Physics, Volume 287 (2009), pp. 1071-1108 | MR | Zbl

[21] Coates, T.; Corti, A.; Iritani, H.; Tseng, H.-H. Computing Genus-Zero Twisted Gromov–Witten Invariants, Duke Math., Volume 147 (2009), pp. 377-438 | MR | Zbl

[22] Coates, T.; Corti, A.; Lee, Y.-P.; Tseng, H.-H. The quantum orbifold cohomology of weighted projective spaces, Acta Math., Volume 202 (2009) no. 2, pp. 139-193 | MR | Zbl

[23] Coates, T.; Givental, A. Quantum Riemann–Roch, Lefschetz and Serre, Annals of mathematics, Volume 165 (2007) no. 1, pp. 15-53 | MR | Zbl

[24] Coates, T.; Iritani, H.; Tseng, H.-H. Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples, Geometry and Topology, Volume 13 (2009) no. 2, pp. 2675-2744 | MR | Zbl

[25] Coates, T.; Ruan, Y. Quantum Cohomology and Crepant Resolutions: A Conjecture (preprint: arXiv:0710.5901)

[26] Deligne, P.; Greene, B.; Yau, S. T. Local behavior of Hodge structures at infinity, Mirror Symmetry II, AMS and International Press, 1997, pp. 683-699 | MR | Zbl

[27] Dolgachev, I. Weighted projective varieties, Proc. Vancouver 1981 (Lecture Notes in Math.), Volume 956, Springer, 1982, pp. 34-71 | MR | Zbl

[28] Faber, C.; Shadrin, S.; Zvonkine, D. Tautological relations and the r-spin Witten conjecture, Annales Scientifiques de l’ENS, Volume 43 (2010) no. 4, pp. 621-658 (preprint version: math.AG/0612510) | Numdam | MR | Zbl

[29] Fan, H.; Jarvis, T.; Merrell, E.; Ruan, Y. Witten’s D 4 Integrable Hierarchies Conjecture (preprint: arXiv:1008.0927)

[30] Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation and its virtual fundamental cycle (preprint: arXiv:0712.4025)

[31] Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation, mirror symmetry and quantum singularity theory (preprint: arXiv:0712.4021v1)

[32] Fan, H.; Jarvis, T.; Ruan, Y. Geometry and analysis of spin equations, Comm. Pure Appl. Math., Volume 61 (2008) no. 6, pp. 745-788 | MR | Zbl

[33] Givental, A. A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996) (Progr. Math.), Volume 160, pp. 141-175 | Zbl

[34] Givental, A. Gromov–Witten invariants and quantization of quadratic hamiltonians, Frobenius manifolds (Aspects Math., E36), Vieweg, Wiesbaden, 2004, pp. 91-112 (preprint version: math.AG/0108100)

[35] Greene, B. R.; Morrison, D. R.; Plesser, M. R. Mirror manifolds in higher dimension, Comm. Math. Phys., Volume 173 (1995) no. 3, pp. 559-597 | MR | Zbl

[36] Herbst, M.; Hori, K.; Page, D. Phases Of N=2 Theories In 1+1 Dimensions With Boundary (DESY-07-154, CERN-PH-TH/2008-048 Preprint version: arXiv:0803.2045)

[37] Hertling, C. tt * geometry, Frobenius manifolds, their connections and their construction for singularities, J. Reine Angew. Math., Volume 555 (2003), pp. 77-161 | MR | Zbl

[38] Hori, K.; Walcher, J. D-branes from matrix factorizations, Strings 04. Part I. C. R. Phys., Volume 5 (2004) no. 9-10, pp. 1061-1070 | MR

[39] Horja, P. Hypergeometric functions and mirror symmetry in toric varieties (preprint: arXiv:math/9912109)

[40] Huang, M.; 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 (arXiv:hep-th/0612125) | MR | Zbl

[41] Intriligator, K.; Vafa, C. Landau–Ginzburg orbifolds, Nuclear Phys. B, Volume 339 (1990) no. 1, pp. 95-120 | MR

[42] Iritani, I. An integral structure in quantum cohomology and mirror symmetry for orbifolds, Adv. in Math., Volume 222 (2009), pp. 1016-1079 (preprint version: arXiv:0903.1463v1) | MR | Zbl

[43] Isik, M. U. Equivalence of the derived category of a variety with a singularity category (preprint: arXiv:1011.1484)

[44] Jarvis, T. J.; Kimura, T.; Vaintrob, A. Moduli spaces of higher spin curves and integrable hierarchies, Compositio Math., Volume 126 (2001) no. 2, pp. 157-212 (math.AG/9905034) | MR | Zbl

[45] Kaufmann, R. A note on the two approaches to stringy functors for orbifolds (preprint: arXiv:math/0703209)

[46] Kaufmann, R. Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry, Contemp. Math., Volume 403 (2006), pp. 67-116 | MR | Zbl

[47] Kontsevich, M. (unpublished)

[48] Kontsevich, M. Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., Volume 147 (1992) no. 1, pp. 1-23 | Zbl

[49] Kontsevich, M.; Manin, Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys., Volume 164 (1994), pp. 525-562 | MR | Zbl

[50] Krawitz, M. FJRW rings and Landau–Ginzburg Mirror Symmetry (preprint: arXiv:0906.0796)

[51] Krawitz, M.; Shen, Y. Landau-Ginzburg/Calabi-Yau correspondence of all genera for elliptic orbifold P 1 (preprint: arXiv:1106.6270)

[52] Krawitz, Marc; Priddis, Nathan; Acosta, Pedro; Bergin, Natalie; Rathnakumara, Himal FJRW-rings and Mirror Symmetry, Comm. Math. Phys., Volume 296 (2010), pp. 145-174 | MR | Zbl

[53] Kreuzer, M.; Skarke, H. On the classification of quasihomogeneous functions, Comm. Math. Phys., Volume 150 (1992) no. 1, pp. 137-147 | MR | Zbl

[54] Kreuzer, M.; Skarke, H. All abelian symmetries of Landau-Ginzburg potentials, Nucl. Phys. B, Volume 405 (1993) no. 2-3, pp. 305-325 (preprint: hep-th/9211047) | MR | Zbl

[55] Li, A.; Ruan, Y. Symplectic surgeries and Gromov–Witten invariants of Calabi–Yau three-folds, Invent. Math., Volume 145 (2001), pp. 151-218 | MR | Zbl

[56] Lian, B.; Liu, K.; Yau, S. Mirror principle. I., Asian J. Math., Volume 1 (1997) no. 4, pp. 729-763 | MR | Zbl

[57] Looijenga, E. J. N. Isolated singular points on complete intersections, London Math. Soc. Lecture Note Series, 77, Cambridge University Press, 1984 | MR

[58] Maulik, D.; Pandharipande, R. A topological view of Gromov–Witten theory, Topology, Volume 45 (2006) no. 5, pp. 887-918 | MR | Zbl

[59] Milanov, T.; Ruan, Y. Gromov–Witten theory of elliptic orbifold P 1 and quasi-modular forms (preprint: arXiv:1106.2321)

[60] Morrison, D. R. Beyond the Kähler cone, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) (Lecture Notes in Phys.), Volume 9, Israel Math. Conf. Proc., pp. 361-376 | Zbl

[61] Morrison, D. R.; Kollár, J. Mathematical Aspects of Mirror Symmetry, Complex Algebraic Geometry (IAS/Park City Math. Series), Volume 3, 1997, pp. 265-340 | MR | Zbl

[62] Orlik, P.; Solomon, L. Singularities II; Automorphisms of forms, Math. Ann., Volume 231 (1978), pp. 229-240 | MR | Zbl

[63] Orlov, D. Derived categories of coherent sheaves and triangulated categories of singularities (preprint: math.AG/0503632) | Zbl

[64] Pham, F. La descente des cols par les onglets de Lefschetz, avec vues sur Gauss–Manin, Systèmes différentiels et singularités (Asterisques), Volume 130, 1985, pp. 11-47 | Numdam | MR | Zbl

[65] Polishchuk, A. Witten’s top Chern class on the moduli space of higher spin curves, Frobenius manifolds (Aspects Math., E36), Vieweg, Wiesbaden, 2004, pp. 253-264 (preprint version: math.AG/0208112) | MR | Zbl

[66] Polishchuk, A.; Vaintrob, A. Chern (preprint: math.AG/0011032)

[67] Polishchuk, A.; Vaintrob, A. Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (Contemp. Math.), Volume 276, Amer. Math. Soc., Providence, RI, 2001, pp. 229-249 (preprint version: math.AG/0011032) | MR | Zbl

[68] Ruan, Y. The Witten equation and geometry of Landau–Ginzburg model (in preparation)

[69] Steenbrink, J. Intersection form for quasi-homogeneous singularities, Compositio Mathematica, Volume 34 (1977) no. 2, pp. 211-223 | Numdam | MR | Zbl

[70] Vafa, C.; Warner, N. Catastrophes and the classification of conformal field theories, Phys. Lett. B, Volume 218 (1989) no. 22, pp. 51 | MR

[71] Wall, C. T. C. A note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980) no. 3, pp. 169-175 | MR | Zbl

[72] Witten, E. Two-dimensional gravity and intersection theory on the moduli space, Surveys in Diff. Geom., Volume 1 (1991), pp. 243-310 | MR | Zbl

[73] Witten, E. Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 235-269 | MR | Zbl

[74] Witten, E. Phases of N=2 theories in two dimensions, Nucl.Phys. B, Volume 403 (1993), pp. 159-222 | MR | Zbl

[75] Zinger, A. Standard vs. reduced genus-one Gromov–Witten invariants, Geom. Topol., Volume 12 (2008) no. 2, pp. 1203-1241 | MR | Zbl

Cité par Sources :