Skip to main content
SpringerLink
Log in

Topological recursion for the conifold transition of a torus knot

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

In this paper we prove a mirror symmetry conjecture based on the work of Brini et al. (Ann Henri Poincaŕe 13(8):1873–1910, 2012) and Diaconescu et al. (Commun Math Phys 319(3):813–863, 2013). This conjecture relates open Gromov–Witten invariants of the conifold transition of a torus knot to the topological recursion on the B-model spectral curve.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. For a comprehensive review, see e.g. [30, 31].

  2. The construction also works for links—we restrict ourselves to knots in this paper.

  3. The equivalence of non-colored HOMFLY and open GW for torus knots is established in [7].

  4. Usually one says the mirror curve to \(\mathcal {X}\) is just \(C_q\). This implicitly considers \(\mathcal {X}\) with Aganagic–Vafa branes (conifold transitions of an unknot) while choosing U, V as the holomorphic functions.

References

  1. Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. arXiv:hep-th/0012041

  2. Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57(1–2), 1–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borot, G., Eynard, B., Orantin, N.: Abstract loop equations, topological recursion, and applications. Commun. Number Theory Phys. 9(1), 51–187 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Remodeling the B-model. Commun. Math. Phys. 287(1), 117–178 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brini, A., Eynard, B., Mariño, M.: Torus knots and mirror symmetry. Ann. Henri Poincaré 13(8), 1873–1910 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cox, D., Katz, S.: Mirror symmetry and algebraic geometry, Math Surveys and monographs, vol. 68. American Math Soc, Providence, RI (1999)

    Book  MATH  Google Scholar 

  7. Diaconescu, D.-E., Shende, V., Vafa, C.: Large N duality, Lagrangian cycles, and algebraic knots. Commun. Math. Phys. 319(3), 813–863 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3(5), 1415–1443 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dunin-Barkowski, P., Orantin, N., Shadrin, S., Spitz, L.: Identification of the Givental formula with the spectral curve topological recursion procedure. Commun. Math. Phys. 328(2), 669–700 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eynard, B.: Intersection number of spectral curves. arXiv:1104.0176

  11. Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eynard, B., Orantin, N.: Computation of open Gromov–Witten invariants for toric Calabi–Yau threefolds by topological recursion, a proof of the BKMP conjecture. Commun. Math. Phys. 337(2), 483–567 (2015)

    Article  MATH  Google Scholar 

  13. Fang, B., Liu, C.-C. M., Zong, Z.: On the remodeling conjecture for toric Calabi–Yau 3-orbifolds. arXiv:1604.07123

  14. Givental, A.: Equivariant Gromov–Witten invariants. Int. Math. Res. Not. 13, 613–663 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Givental, A.: A Mirror Theorem for Toric Complete Intersections, Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progress in Mathematics, vol. 160, pp. 141–175. Birkhäuser, Boston, MA (1998)

    Google Scholar 

  16. Givental, A.: Elliptic Gromov–Witten Invariants and the Generalized Mirror Conjecture, Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), pp. 107–155. World Scientific Publishing, River Edge, NJ (1998)

    MATH  Google Scholar 

  17. Givental, A.: Semisimple Frobenius structures at higher genus. Int. Math. Res. Not. 23, 1265–1286 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Givental, A.: Gromov–Witten invariants and quantization of quadratic hamiltonians. Mosc. Math. J. 1(4), 551–568 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gu, J., Jockers, H., Klemm, A., Soroush, M.: Knot invariants from topological recursion on augmentation varieties. Commun. Math. Phys. 336(2), 987–1051 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Katz, S., Liu, C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5(1), 1–49 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Koshkin, S.: Conormal bundles to knots and the Gopakumar–Vafa conjecture. Adv. Theor. Math. Phys. 11(4), 591–634 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Labastida, J.M.F., Mariño, M., Vafa, C.: Knots, links and branes at large N. J. High Energy Phys. 11, 42 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Li, J., Liu, C.-C., Liu, K., Zhou, J.: A mathematical theory of the topological vertex. Geom. Topol. 13(1), 527–621 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lee, Y.-P., Pandharipande, R.: Frobenius manifolds, Gromov–Witten theory and virasoro constraints (preprint). http://www.math.utah.edu/~yplee/research/#publications

  25. Lian, B., Liu, K., Yau, S.-T.: Mirror principle I. Asian J. Math. 1(4), 729–763 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lian, B., Liu, K., Yau, S.-T.: Mirror principle II. Asian J. Math. 3(1), 109–146 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, C.-C. M.: Moduli of J-holomorphic curves with Lagrangian boundary conditions and open Gromov–Witten invariants for an \(S^1\)-equivariant pair. arXiv:math/0210257

  28. Liu, C.-C.M., Liu, K., Zhou, J.: A proof of a conjecture of Mario–Vafa on Hodge integrals. J. Differ. Geom. 65(2), 289–340 (2003)

    Article  MATH  Google Scholar 

  29. Mahowald, M.: Knots and Gamma classes in open topological string theory. Ph.D. thesis, Northwestern University (2016)

  30. Mariño, M.: Chern–Simons theory and topological strings. Rev. Modern Phys. 77(2), 675–720 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mariño, M.: Chern–Simons theory, the 1/N expansion, and string theory. arXiv:1001.2542

  32. Mariño, M., Vafa, C.: Framed Knots at Large N, Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemporary Mathematics, vol. 310, pp. 185–204. American Mathematical Society, Providence, RI (2002)

    Google Scholar 

  33. McDuff, D., Salamon, D.: Introduction to Symplectic Topology, Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, New York (1998)

    MATH  Google Scholar 

  34. Okounkov, A., Pandharipande, R.: Hodge integrals and invariants of the unknot. Geom. Topol. 8, 675–699 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ooguri, H., Vafa, C.: Knot invariants and topological strings. Nucl. Phys. B. 577(3), 419–438 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Taubes, C.H.: Lagrangians for the Gopakumar–Vafa conjecture. Adv. Theor. Math. Phys. 1, 139–163 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Witten, E.: Chern-Simons Gauge Theory as a String Theory, The Floer Memorial Volume, Progress in Mathematics, vol. 133, pp. 637–678. Birkhäuser, Basel (1995)

    MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Chiu-Chu Melissa Liu for very helpful discussion and the wonderful collaboration in many related projects—those projects are indispensable to this one. We also thank her for the construction of disk invariants using relative Gromov–Witten invariants in our case. We would like to thank the anonymous referee for the valuable suggestion. The first author would like to thank Sergei Gukov for the discussion on the localization computation for torus knots. The work of BF is partially supported by a start-up Grant at Peking University. The work of ZZ is partially supported by the start-up Grant at Tsinghua University.

Author information

Authors and Affiliations

  1. Beijing International Center for Mathematical Research, Peking University, 5 Yiheyuan Road, Beijing, 100871, China

    Bohan Fang

  2. Yau Mathematical Sciences Center, Tsinghua University, Jin Chun Yuan West Building Room 138, Haidian District, Beijing, 100084, China

    Zhengyu Zong

Authors
  1. Bohan Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhengyu Zong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhengyu Zong.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fang, B., Zong, Z. Topological recursion for the conifold transition of a torus knot. Sel. Math. New Ser. 25, 35 (2019). https://doi.org/10.1007/s00029-019-0483-x

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s00029-019-0483-x

Mathematics Subject Classification

Search

Navigation