## Duke Mathematical Journal

### Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds

#### Abstract

Let $X$ be a compact toric Kähler manifold with $-K_{X}$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya, Oh, Ohta, and Ono defined open Gromov–Witten (GW) invariants of $X$ as virtual counts of holomorphic disks with Lagrangian boundary condition $L$. We prove a formula that equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\mathbb{P}^{1}$ used by Seidel and McDuff earlier to construct Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disk potential of $X$, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya, Oh, Ohta, and Ono.

#### Article information

Source
Duke Math. J., Volume 166, Number 8 (2017), 1405-1462.

Dates
Received: 13 July 2015
Revised: 29 June 2016
First available in Project Euclid: 25 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1487991810

Digital Object Identifier
doi:10.1215/00127094-0000003X

Mathematical Reviews number (MathSciNet)
MR3659939

Zentralblatt MATH identifier
1371.53090

#### Citation

Chan, Kwokwai; Lau, Siu-Cheong; Leung, Naichung Conan; Tseng, Hsian-Hua. Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds. Duke Math. J. 166 (2017), no. 8, 1405--1462. doi:10.1215/00127094-0000003X. https://projecteuclid.org/euclid.dmj/1487991810

#### References

• [1] D. Auroux, Mirror symmetry and $T$-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51–91.
• [2] D. Auroux, “Special Lagrangian fibrations, wall-crossing, and mirror symmetry” in Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry, Surv. Differ. Geom. 13, Int. Press, Somerville, Mass., 2009, 1–47.
• [3] V. V. Batyrev, Quantum cohomology rings of toric manifolds, Astérisque 218 (1993), 9–34.
• [4] K. Chan, A formula equating open and closed Gromov–Witten invariants and its applications to mirror symmetry, Pacific J. Math. 254 (2011), 275–293.
• [5] K. Chan and S.-C. Lau, Open Gromov–Witten invariants and superpotentials for semi-Fano toric surfaces, Int. Math. Res. Not. IMRN 2014, no. 14, 3759–3789.
• [6] K. Chan, S.-C. Lau, and N. C. Leung, SYZ mirror symmetry for toric Calabi–Yau manifolds, J. Differential Geom. 90 (2012), 177–250.
• [7] K. Chan, S.-C. Lau, and H.-H. Tseng, Enumerative meaning of mirror maps for toric Calabi–Yau manifolds, Adv. Math. 244 (2013), 605–625.
• [8] C.-H. Cho and Y.-G. Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), 773–814.
• [9] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I, AMS/IP Stud. Adv. Math. 46.1, Amer. Math. Soc., Providence, 2009.
• [10] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part II, AMS/IP Stud. Adv. Math. 46.2, Amer. Math. Soc., Providence, 2009.
• [11] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151 (2010), 23–174.
• [12] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory on compact toric manifolds, II: Bulk deformations, Selecta Math. (N.S.) 17 (2011), 609–711.
• [13] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Toric degeneration and nondisplaceable Lagrangian tori in $S^{2}\times S^{2}$, Int. Math. Res. Not. IMRN 2012, no. 13, 2942–2993.
• [14] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory and mirror symmetry on compact toric manifolds, Astérisque 376, Soc. Math. France, Paris, 2016.
• [15] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory, preprint, arXiv:1105.5123v2 [math.SG].
• [16] W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, 1993.
• [17] A. Givental, “A mirror theorem for toric complete intersections” in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math. 160, Birkhäuser, Boston, 1998, 141–175.
• [18] E. González and H. Iritani, Seidel elements and mirror transformations, Selecta Math. (N.S.) 18 (2012), 557–590.
• [19] E. González and H. Iritani, Seidel elements and potential functions of holomorphic disc counting, preprint, arXiv:1301.5454v2 [math.SG].
• [20] K. Hori and C. Vafa, Mirror symmetry, preprint, arXiv:hep-th/0002222v3.
• [21] H. Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), 1016–1079.
• [22] S.-C. Lau, N. C. Leung, and B. Wu, A relation for Gromov–Witten invariants of local Calabi–Yau threefolds, Math. Res. Lett. 18 (2011), 943–956.
• [23] S.-C. Lau, N. C. Leung, and B. Wu, Mirror maps equal SYZ maps for toric Calabi–Yau surfaces, Bull. Lond. Math. Soc. 44 (2012), 255–270.
• [24] W. Lerche and P. Mayr, On $\mathscr{N}=1$ mirror symmetry for open Type II strings, preprint, arXiv:hep-th/0111113v2.
• [25] W. Lerche, P. Mayr, and N. Warner, $\mathscr{N}=1$ special geometry, mixed Hodge variations and toric geometry, preprint, arXiv:hep-th/0208039v1.
• [26] B. Lian, K. Liu, and S.-T. Yau, Mirror principle, III, Asian J. Math. 3 (1999), 771–800.
• [27] P. Mayr, $\mathscr{N}=1$ mirror symmetry and open/closed string duality, Adv. Theor. Math. Phys. 5 (2001), 213–242.
• [28] P. Mayr, Summing up open string instantons and $\mathscr{N}=1$ string amplitudes, preprint, arXiv:hep-th/0203237v2.
• [29] D. McDuff, Quantum homology of fibrations over $S^{2}$, Internat. J. Math. 11 (2000), 665–721.
• [30] D. McDuff and S. Tolman, Topological properties of Hamiltonian circle actions, IMRP Int. Math. Res. Pap. 2006, no. 72826, 1–77.
• [31] P. Seidel, $\pi_{1}$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), 1046–1095.