Landau-Ginzburg/Calabi-Yau correspondence via wall-crossing

Abstract

Landau-Ginzburg/Calabi-Yau correspondence claims the equivalence between the Gromov-Witten theory and the Fan-Jarvis-Ruan-Witten theory. The authors survey recently developed wall-crossing approach to the Landau-Ginzburg/Calabi-Yau correspondence for a quintic threefold.

Keywords

Gromov-Witten invariant / Fan-Jarvis-Ruan-Witten invariant / Moduli space / Quasimap / Wall-crossing

References

[1] Behrend K., Fantechi B.. The intrinsic normal cone. Invent. Math., 1997, 129(1): 45-88

[2] Chang, H.-L., Kiem, Y.-H. and Li, J., Torus localization and wall crossing formulas for cosection localized virtual cycles, arXiv: 1502.00078.

[3] Chang H.-L., Li J.. Gromov-Witten invariants of stable maps with fields. Int. Math. Res. Notices, 2012, 18: 4163-4217

[4] Chang H.-L., Li J., Li W.-P.. Wittens top Chern class via cosection localization. Invent. Math., 2015, 200(3): 1015-1063

[5] Cheong D., Ciocan-Fontanine I., Kim B.. Orbifold quasimap theory. Math. Ann., 2015, 363(3): 777-816

[6] Chiodo A.. Stable twisted curves and their r-spin structures. Ann. Inst. Fourier, 2008, 58(5): 1635-1689

[7] Chiodo A., Ruan Y.. Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations. Invent. Math., 2010, 182(1): 117-165

[8] Choi, J. and Kiem, Y.-H., Landau-Ginzburg/Calabi-Yau correspondence via quasi-maps, I, arXiv: 1103. 0833.

[9] Ciocan-Fontanine, I. and Kim, B., Wall-crossing in genus zero quasimap theory and mirror maps, arXiv: 1304.7056.

[10] Ciocan-Fontanine, I. and Kim, B., Higher genus quasimap wall-crossing for semi-positive targets, arXiv: 1308.6377.

[11] Ciocan-Fontanine I., Kim B., Maulik D.. Stable quasimaps to GIT quotients. J. Geom. and Phys., 2014, 75: 17-47

[12] Clader, E., Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections X _3,3 and X _2,2,2,2, arXiv: 1301.5530.

[13] Cotterill E.. Rational curves of degree 10 on a general quintic threefold. Comm. Alg., 2005, 33(6): 1833-1872

[14] Cotterill E.. Rational curves of degree 11 on a general quintic 3-fold. Quart. J. Math., 2012, 63(3): 539-568

[15] Cox D., Katz S.. Mirror symmetry and algebraic geometry. Mathematical surveys and monographs, 1999

[16] Faltings G.. Moduli-stacks for bundles on semistable curves. Math. Ann., 1996, 304: 489-515

[17] Fan H., Jarvis T., Ruan Y.. The Witten equation, mirror symmetry, and quantum singularity theory. Ann. Math., 2013, 178: 1-106

[18] Fan, H., Jarvis, T. and Ruan, Y., A mathematical theory of the gauged linear sigma model, arXiv: 1506.02109.

[19] Fulton W.. Intersection Theory, 1998 second

[20] Fulton W., MacPherson R.. A compactification of configuration spaces. Ann. Math., 1994, 139(1): 183-225

[21] Hori K., Katz S., Klemm A., Pandharipande R. Mirror Symmetry, 2003

[22] Jarvis T., Kimura T., Vaintrob A.. Moduli spaces of higher spin curves and integrable hierarchies. Comp. Math., 2001, 126: 157-212

[23] Johnsen T., Kleiman S.. Rational curves of degree at most 9 on a general quintic threefold. Comm. Alg., 1996, 24: 2721-2753

[24] Katz S.. On the finiteness of rational curves on quintic threefolds. Comp. Math., 1986, 60: 151-162

[25] Kiem Y.-H., Li J.. Localizing virtual cycles by cosections. J. Amer. Math. Soc., 2013, 26: 1025-1050

[26] Le Potier J.. Systèmes cohérents et structures de niveau, 1993

[27] Li J., Tian G.. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc., 1998, 11: 119-174

[28] Marian, A., Oprea D. and Pandharipande, R., The moduli space of stable quotients. arXiv: 0904.2992.

[29] Ross D., Ruan Y.. Wall-crossing in genus zero Landau-Ginzburg theory. J. Reine Angew. Math., 2015

[30] Toda Y.. Moduli spaces of stable quotients and wall crossing phenomena. Comp. Math., 2011, 147: 1479-1518

[31] Vafa C., Warner N.. Catastrophes and the classification of conformal field theories. Phys. Lett. B, 1989, 218(1): 51-58

[32] Vainsencher I.. Enumeration of n-fold tangent hyperplanes to a surface. J. Algebraic Geom., 1995, 4: 503-526

[33] Witten E.. Phases of N = 2 theories in two dimensions. Nucl. Phys. B, 1993, 403: 159-222