Relative Gromov–Witten invariants and the enumerative meaning of mirror maps for toric Calabi–Yau orbifolds

You, Fenglong

doi:10.1090/tran/8196

Relative Gromov–Witten invariants and the enumerative meaning of mirror maps for toric Calabi–Yau orbifolds
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Abstract:

We provide an enumerative meaning of the mirror maps for toric Calabi–Yau orbifolds in terms of relative Gromov–Witten invariants of the toric compactifications. As a consequence, we obtain an equality between relative Gromov–Witten invariants and open Gromov–Witten invariants. Therefore, the instanton corrected mirrors for toric Calabi–Yau orbifolds can be constructed using relative Gromov–Witten invariants.

References

D. Abramovich, Lectures on Gromov-Witten invariants of orbifolds, Enumerative invariants in algebraic geometry and string theory, Lecture Notes in Math., vol. 1947, Springer, Berlin, 2008, pp. 1–48. MR 2493583, DOI 10.1007/978-3-540-79814-9_{1}
Dan Abramovich, Charles Cadman, and Jonathan Wise, Relative and orbifold Gromov-Witten invariants, Algebr. Geom. 4 (2017), no. 4, 472–500. MR 3683504, DOI 10.14231/ag-2017-025
Dan Abramovich, Tom Graber, and Angelo Vistoli, Algebraic orbifold quantum products, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 1–24. MR 1950940, DOI 10.1090/conm/310/05397
Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. MR 2450211, DOI 10.1353/ajm.0.0017
Denis Auroux, Mirror symmetry and $T$-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51–91. MR 2386535
Denis Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, Surv. Differ. Geom., vol. 13, Int. Press, Somerville, MA, 2009, pp. 1–47. MR 2537081, DOI 10.4310/SDG.2008.v13.n1.a1
Lev A. Borisov, Linda Chen, and Gregory G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no. 1, 193–215. MR 2114820, DOI 10.1090/S0894-0347-04-00471-0
Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. MR 2306040, DOI 10.1353/ajm.2007.0007
Charles Cadman and Linda Chen, Enumeration of rational plane curves tangent to a smooth cubic, Adv. Math. 219 (2008), no. 1, 316–343. MR 2435425, DOI 10.1016/j.aim.2008.04.013
Kwokwai Chan, Cheol-Hyun Cho, Siu-Cheong Lau, and Hsian-Hua Tseng, Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds, J. Differential Geom. 103 (2016), no. 2, 207–288. MR 3504949
Kwokwai Chan, Siu-Cheong Lau, and Naichung Conan Leung, SYZ mirror symmetry for toric Calabi-Yau manifolds, J. Differential Geom. 90 (2012), no. 2, 177–250. MR 2899874
Kwokwai Chan, Siu-Cheong Lau, and Hsian-Hua Tseng, Enumerative meaning of mirror maps for toric Calabi-Yau manifolds, Adv. Math. 244 (2013), 605–625. MR 3077883, DOI 10.1016/j.aim.2013.05.018
Weimin Chen and Yongbin Ruan, Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25–85. MR 1950941, DOI 10.1090/conm/310/05398
Weimin Chen and Yongbin Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004), no. 1, 1–31. MR 2104605, DOI 10.1007/s00220-004-1089-4
Daewoong Cheong, Ionuţ Ciocan-Fontanine, and Bumsig Kim, Orbifold quasimap theory, Math. Ann. 363 (2015), no. 3-4, 777–816. MR 3412343, DOI 10.1007/s00208-015-1186-z
Cheol-Hyun Cho and Mainak Poddar, Holomorphic orbi-discs and Lagrangian Floer cohomology of symplectic toric orbifolds, J. Differential Geom. 98 (2014), no. 1, 21–116. MR 3263515
Cheol-Hyun Cho and Hyung-Seok Shin, Chern-Weil Maslov index and its orbifold analogue, Asian J. Math. 20 (2016), no. 1, 1–19. MR 3460756, DOI 10.4310/AJM.2016.v20.n1.a1
Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912. MR 3414388, DOI 10.1112/S0010437X15007356
Tom Coates, Amin Gholampour, Hiroshi Iritani, Yunfeng Jiang, Paul Johnson, and Cristina Manolache, The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces, Math. Res. Lett. 19 (2012), no. 5, 997–1005. MR 3039825, DOI 10.4310/MRL.2012.v19.n5.a3
Honglu Fan, Hsian-Hua Tseng, and Fenglong You, Mirror theorems for root stacks and relative pairs, Selecta Math. (N.S.) 25 (2019), no. 4, Paper No. 54, 25. MR 3997137, DOI 10.1007/s00029-019-0501-z
H. Fan, L. Wu WDVV equation and its application to relative Gromov–Witten theory. arXiv:1902.05739. To appear in Internat. Math. Res. Notices.
H. Fan, L. Wu, and F. You, Structures in genus-zero relative Gromov–Witten theory, J. Topol. 13 (2020), no. 1, 269–307.
H. Fan, L. Wu, and F. You, Higher genus relative Gromov–Witten theory and Double ramification cycles, arXiv:1907.07133, 2019.
Alexander B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13 (1996), 613–663. MR 1408320, DOI 10.1155/S1073792896000414
Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996) Progr. Math., vol. 160, Birkhäuser Boston, Boston, MA, 1998, pp. 141–175. MR 1653024
Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. MR 2115767
Mark Gross, Paul Hacking, and Sean Keel, Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168. MR 3415066, DOI 10.1007/s10240-015-0073-1
Mark Gross, Rahul Pandharipande, and Bernd Siebert, The tropical vertex, Duke Math. J. 153 (2010), no. 2, 297–362. MR 2667135, DOI 10.1215/00127094-2010-025
Mark Gross and Bernd Siebert, From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), no. 3, 1301–1428. MR 2846484, DOI 10.4007/annals.2011.174.3.1
Mark Gross and Bernd Siebert, Local mirror symmetry in the tropics, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 723–744. MR 3728635
Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, DOI 10.4007/annals.2003.157.45
Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. MR 2553377, DOI 10.1016/j.aim.2009.05.016
Yunfeng Jiang, The orbifold cohomology ring of simplicial toric stack bundles, Illinois J. Math. 52 (2008), no. 2, 493–514. MR 2524648
Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465. MR 3606511, DOI 10.1007/s11005-016-0903-1
Siu-Cheong Lau, Gross-Siebert’s slab functions and open GW invariants for toric Calabi-Yau manifolds, Math. Res. Lett. 22 (2015), no. 3, 881–898. MR 3350109, DOI 10.4310/MRL.2015.v22.n3.a13
An-Min Li and Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218. MR 1839289, DOI 10.1007/s002220100146
Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667
Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
Bong H. Lian, Kefeng Liu, and Shing-Tung Yau, Mirror principle. I, Asian J. Math. 1 (1997), no. 4, 729–763. MR 1621573, DOI 10.4310/AJM.1997.v1.n4.a5
H. Ruddat and B. Siebert, Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations, arXiv:1907.03794, 2019.
Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831, DOI 10.1016/0550-3213(96)00434-8
Hsian-Hua Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre, Geom. Topol. 14 (2010), no. 1, 1–81. MR 2578300, DOI 10.2140/gt.2010.14.1
H.-H. Tseng and F. You, Higher genus relative and orbifold Gromov–Witten invariants of curves, arXiv:1804.09905, 2018.
H.-H. Tseng and F. You, Higher genus relative and orbifold Gromov-Witten invariants, arXiv:1806.11082. Geom. Topol. (to appear).
H.-H. Tseng and F. You, in preparation.