A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence
Annales de l'Institut Fourier, Volume 61 (2011) no. 7, p. 2803-2864
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.
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.
DOI : https://doi.org/10.5802/aif.2795
Classification:  14J33,  14J32,  14H10
Keywords: Mirror symmetry, Gromov–Witten theory, Calabi–Yau varieties, moduli of curves
@article{AIF_2011__61_7_2803_0,
     author = {Chiodo, Alessandro and Ruan, Yongbin},
     title = {A global mirror symmetry framework for the Landau--Ginzburg/Calabi--Yau correspondence},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {7},
     year = {2011},
     pages = {2803-2864},
     doi = {10.5802/aif.2795},
     zbl = {pre06193028},
     mrnumber = {3112509},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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, Volume 61 (2011) no. 7, pp. 2803-2864. doi : 10.5802/aif.2795. https://aif.centre-mersenne.org/item/AIF_2011__61_7_2803_0/

[1] Abramovich, D.; Graber, T.; Vistoli, A. Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math., Tome 130 (2008) no. 5 | MR 2450211 | Zbl 1193.14070

[2] Abramovich, D.; Jarvis, T. J. Moduli of twisted spin curves, Proc. Amer. Math. Soc., Tome 131 (2003), pp. 685-699 (preprint version: math.AG/0104154) | MR 1937405 | Zbl 1037.14008

[3] Aganagic, M.; Bouchard, V.; Klemm, A. Topological Strings and (Almost) Modular Forms, Commun. Math. Phys., Tome 277 (2008), pp. 771-819 (preprint version: hep-th/0607100) | MR 2365453 | Zbl 1165.81037

[4] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps, Birkhäuser, Boston Tome II (1988) | MR 966191

[5] Batyrev, V. V.; Borisov, L. A. Dual Cones and Mirror Symmetry for Generalized Calabi–Yau Manifolds, Mirror Symmetry II, Amer. Math. Soc. Providence, RI (AMS/IP Stud. Adv. Math 1) (1997), pp. 71-86 | MR 1416334 | Zbl 0927.14019

[6] Berglund, P.; Hübsch, T. A Generalized Construction of Mirror Manifolds, Nuclear Physics B, Tome 393 (1993), p. 397-391 | MR 1214325 | Zbl 1245.14039

[7] Berglund, P.; Katz, S. Mirror Symmetry Constructions: A Review, Mirror Symmetry II, Amer. Math. Soc. Providence, RI (AMS/IP Stud. Adv. Math 1) (1997), pp. 87-113 (preprint version: arXiv:hep-th/9406008) | MR 1416335 | Zbl 0919.14023

[8] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C. Kodaira–Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Comm. Math. Phy., Tome 165 (1994), pp. 311-427 | MR 1301851 | Zbl 0815.53082

[9] Boissière, S.; Mann, É; Perroni, F. A model for the orbifold Chow ring of weighted projective spaces, Communications in Algebra, Tome 37 (2009), pp. 503-514 | MR 2493797 | Zbl 1178.14056

[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, Tome 359 (1991), pp. 21-74 | MR 1115626 | Zbl 1098.32506

[12] Chiodo, A. The Witten top Chern class via K-theory, J. Algebraic Geom., Tome 15 (2006) no. 4, pp. 681-707 (preprint version: math.AG/0210398) | MR 2237266 | Zbl 1117.14008

[13] Chiodo, A. Stable twisted curves and their r-spin structures (Courbes champêtres stables et leurs structures r-spin), Ann. Inst. Fourier, Tome 58 (2008) no. 5, pp. 1635-1689 (preprint version: math.AG/0603687) | Numdam | MR 2445829 | Zbl 1179.14028

[14] Chiodo, A. Towards an enumerative geometry of the moduli space of twisted curves and rth roots, Compos. Math., Tome 144 (2008), pp. 1461-1496 (Part 6, preprint version: math.AG/0607324) | MR 2474317 | Zbl 1166.14018

[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., Tome 182 (2010), pp. 117-165 (preprint version: arXiv:0812.4660) | MR 2672282 | Zbl 1197.14043

[17] Chiodo, A.; Ruan, Y. LG/CY correspondence: the state space isomorphism, Adv. Math., Tome 227 (2011) no. 6, pp. 2157-2188 | MR 2807086 | Zbl 1245.14038

[18] Chiodo, A.; Zvonkine, D. Twisted Gromov–Witten r-spin potentials and Givental’s quantization, Adv. Theor. Math. Phys., Tome 13 (2009) no. 5, pp. 1335-1369 (preprint version: arXiv:0711.0339) | MR 2672465 | Zbl 1204.81099

[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, Tome 287 (2009), pp. 1071-1108 | MR 2486673 | Zbl 1200.53081

[21] Coates, T.; Corti, A.; Iritani, H.; Tseng, H.-H. Computing Genus-Zero Twisted Gromov–Witten Invariants, Duke Math., Tome 147 (2009), pp. 377-438 | MR 2510741 | Zbl 1176.14009

[22] Coates, T.; Corti, A.; Lee, Y.-P.; Tseng, H.-H. The quantum orbifold cohomology of weighted projective spaces, Acta Math., Tome 202 (2009) no. 2, pp. 139-193 | MR 2506749 | Zbl 1213.53106

[23] Coates, T.; Givental, A. Quantum Riemann–Roch, Lefschetz and Serre, Annals of mathematics, Tome 165 (2007) no. 1, pp. 15-53 | MR 2276766 | Zbl 1189.14063

[24] Coates, T.; Iritani, H.; Tseng, H.-H. Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples, Geometry and Topology, Tome 13 (2009) no. 2, pp. 2675-2744 | MR 2529944 | Zbl 1184.53086

[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 1416353 | Zbl 0939.14005

[27] Dolgachev, I. Weighted projective varieties, Proc. Vancouver 1981, Springer (Lecture Notes in Math.) Tome 956 (1982), pp. 34-71 | MR 704986 | Zbl 0516.14014

[28] Faber, C.; Shadrin, S.; Zvonkine, D. Tautological relations and the r-spin Witten conjecture, Annales Scientifiques de l’ENS, Tome 43 (2010) no. 4, pp. 621-658 (preprint version: math.AG/0612510) | Numdam | MR 2722511 | Zbl 1203.53090

[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., Tome 61 (2008) no. 6, pp. 745-788 | MR 2400605 | Zbl 1141.58012

[33] Givental, A. A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996) (Progr. Math.) Tome 160, pp. 141-175 | Zbl 0936.14031

[34] Givental, A. Gromov–Witten invariants and quantization of quadratic hamiltonians, Frobenius manifolds, Vieweg, Wiesbaden (Aspects Math., E36) (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., Tome 173 (1995) no. 3, pp. 559-597 | MR 1357989 | Zbl 0842.32014

[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., Tome 555 (2003), pp. 77-161 | MR 1956595 | Zbl 1040.53095

[38] Hori, K.; Walcher, J. D-branes from matrix factorizations, Strings 04. Part I. C. R. Phys., Tome 5 (2004) no. 9-10, pp. 1061-1070 | MR 2121690

[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, Springer, Berlin (Lecture Notes in Phys.) Tome 757 (2009), pp. 45-102 (arXiv:hep-th/0612125) | MR 2596635 | Zbl 1166.81358

[41] Intriligator, K.; Vafa, C. Landau–Ginzburg orbifolds, Nuclear Phys. B, Tome 339 (1990) no. 1, pp. 95-120 | MR 1061738

[42] Iritani, I. An integral structure in quantum cohomology and mirror symmetry for orbifolds, Adv. in Math., Tome 222 (2009), pp. 1016-1079 (preprint version: arXiv:0903.1463v1) | MR 2553377 | Zbl 1190.14054

[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., Tome 126 (2001) no. 2, pp. 157-212 (math.AG/9905034) | MR 1827643 | Zbl 1015.14028

[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., Tome 403 (2006), pp. 67-116 | MR 2234885 | Zbl 1116.14037

[47] Kontsevich, M. (unpublished)

[48] Kontsevich, M. Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., Tome 147 (1992) no. 1, pp. 1-23 | Zbl 0756.35081

[49] Kontsevich, M.; Manin, Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys., Tome 164 (1994), pp. 525-562 | MR 1291244 | Zbl 0853.14020

[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., Tome 296 (2010), pp. 145-174 | MR 2606631 | Zbl 1250.81087

[53] Kreuzer, M.; Skarke, H. On the classification of quasihomogeneous functions, Comm. Math. Phys., Tome 150 (1992) no. 1, pp. 137-147 | MR 1188500 | Zbl 0767.57019

[54] Kreuzer, M.; Skarke, H. All abelian symmetries of Landau-Ginzburg potentials, Nucl. Phys. B, Tome 405 (1993) no. 2-3, pp. 305-325 (preprint: hep-th/9211047) | MR 1240688 | Zbl 0990.81635

[55] Li, A.; Ruan, Y. Symplectic surgeries and Gromov–Witten invariants of Calabi–Yau three-folds, Invent. Math., Tome 145 (2001), pp. 151-218 | MR 1839289 | Zbl 1062.53073

[56] Lian, B.; Liu, K.; Yau, S. Mirror principle. I., Asian J. Math., Tome 1 (1997) no. 4, pp. 729-763 | MR 1621573 | Zbl 0953.14026

[57] Looijenga, E. J. N. Isolated singular points on complete intersections, Cambridge University Press, London Math. Soc. Lecture Note Series, Tome 77 (1984) | MR 747303 | Zbl pre06152219

[58] Maulik, D.; Pandharipande, R. A topological view of Gromov–Witten theory, Topology, Tome 45 (2006) no. 5, pp. 887-918 | MR 2248516 | Zbl 1112.14065

[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), Israel Math. Conf. Proc. (Lecture Notes in Phys.) Tome 9, pp. 361-376 | Zbl 0847.32034

[61] Morrison, D. R.; Kollár, J. Mathematical Aspects of Mirror Symmetry, Complex Algebraic Geometry (IAS/Park City Math. Series) Tome 3 (1997), pp. 265-340 | MR 1442525 | Zbl 0932.14022

[62] Orlik, P.; Solomon, L. Singularities II; Automorphisms of forms, Math. Ann., Tome 231 (1978), pp. 229-240 | MR 476735 | Zbl 0352.14002

[63] Orlov, D. Derived categories of coherent sheaves and triangulated categories of singularities (preprint: math.AG/0503632) | Zbl 0996.18007

[64] Pham, F. La descente des cols par les onglets de Lefschetz, avec vues sur Gauss–Manin, Systèmes différentiels et singularités (Asterisques) Tome 130 (1985), pp. 11-47 | MR 804048 | Zbl 0597.32012

[65] Polishchuk, A. Witten’s top Chern class on the moduli space of higher spin curves, Frobenius manifolds, Vieweg, Wiesbaden (Aspects Math., E36) (2004), pp. 253-264 (preprint version: math.AG/0208112) | MR 2115773 | Zbl 1105.14010

[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), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 276 (2001), pp. 229-249 (preprint version: math.AG/0011032) | MR 1837120 | Zbl 1051.14007

[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, Tome 34 (1977) no. 2, pp. 211-223 | Numdam | MR 453735 | Zbl 0347.14001

[70] Vafa, C.; Warner, N. Catastrophes and the classification of conformal field theories, Phys. Lett. B, Tome 218 (1989) no. 22, pp. 51 | MR 983349

[71] Wall, C. T. C. A note on symmetry of singularities, Bull. London Math. Soc., Tome 12 (1980) no. 3, pp. 169-175 | MR 572095 | Zbl 0427.32010

[72] Witten, E. Two-dimensional gravity and intersection theory on the moduli space, Surveys in Diff. Geom., Tome 1 (1991), pp. 243-310 | MR 1144529 | Zbl 0757.53049

[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 1215968 | Zbl 0812.14017

[74] Witten, E. Phases of N=2 theories in two dimensions, Nucl.Phys. B, Tome 403 (1993), pp. 159-222 | MR 1232617 | Zbl 0910.14020

[75] Zinger, A. Standard vs. reduced genus-one Gromov–Witten invariants, Geom. Topol., Tome 12 (2008) no. 2, pp. 1203-1241 | MR 2403808 | Zbl 1167.14009