Skip to main content
SpringerLink
Log in

Gopakumar–Vafa Type Invariants on Calabi–Yau 4-Folds via Descendent Insertions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The Gopakumar–Vafa type invariants on Calabi–Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm–Pandharipande from Gromov–Witten theory, and their integrality is conjectured. In a previous work of Cao–Maulik–Toda, \(\mathop {\mathrm{DT}}\nolimits _4\) invariants with primary insertions on moduli spaces of one dimensional stable sheaves are used to give a sheaf theoretical interpretation of the genus zero GV type invariants. In this paper, we propose a sheaf theoretical interpretation of the genus one GV type invariants using descendent insertions on the above moduli spaces. The conjectural formula in particular implies nontrivial constraints on genus zero GV type (equivalently GW) invariants of CY 4-folds which can be proved by the WDVV equation.

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

Similar content being viewed by others

Notes

  1. In this paper, a Calabi–Yau 4-fold is a complex smooth projective 4-fold X satisfying \(K_X \cong \mathcal {O}_X\).

  2. They are integers because of Ionel–Parker’s proof of genus zero integrality [IP, Theorem 9.2].

  3. We are very grateful to Sergej Monavari for sharing his computation to us [Monavari].

References

  1. Borisov, D., Joyce, D.: Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds. Geom. Topol. 21, 3231–3311 (2017)

    Article  MathSciNet  Google Scholar 

  2. Cao, Y.: Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds II: Fano 3-folds. Commun. Contemp. Math. 22(7), 1950060 (2020)

    Article  MathSciNet  Google Scholar 

  3. Cao, Y., Gross, J., Joyce, D.: Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds. Adv. Math. 368, 107134 (2020)

    Article  MathSciNet  Google Scholar 

  4. Cao, Y., Kool, M.: Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds. Adv. Math. 338, 601–648 (2018)

    Article  MathSciNet  Google Scholar 

  5. Cao, Y., Kool, M.: Curve counting and DT/PT correspondence for Calabi–Yau 4-folds. Adv. Math. 375, 107371 (2020)

    Article  MathSciNet  Google Scholar 

  6. Cao, Y., Kool, M., Monavari, S.: K-theoretic DT/PT correspondence for toric Calabi–Yau 4-folds. arXiv:1906.07856

  7. Cao, Y., Kool, M., Monavari, S.: Stable pair invariants of local Calabi–Yau 4-folds. arXiv:2004.09355

  8. Cao, Y., Leung, N.C.: Donaldson–Thomas theory for Calabi–Yau 4-folds. arXiv:1407.7659

  9. Cao, Y., Leung, N.C.: Orientability for gauge theories on Calabi–Yau manifolds. Adv. Math. 314, 48–70 (2017)

    Article  MathSciNet  Google Scholar 

  10. Cao, Y., Maulik, D., Toda, Y.: Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds. Adv. Math. 338, 41–92 (2018)

    Article  MathSciNet  Google Scholar 

  11. Cao, Y., Maulik, D., Toda, Y.: Stable pairs and Gopakumar–Vafa type invariants for Calabi–Yau 4-folds. J. Eur. Math. Soc. (JEMS) (to appear). arXiv:1902.00003

  12. Cao, Y., Toda, Y.: Curve counting via stable objects in derived categories of Calabi–Yau 4-folds. arXiv:1909.04897

  13. Cao, Y., Toda, Y.: Tautological stable pair invariants of Calabi–Yau 4-folds. arXiv:2009.03553

  14. Cao, Y., Toda, Y.: Counting perverse coherent systems on Calabi–Yau 4-folds. arXiv:2009.10909

  15. Cox, D.A., Katz, S.: Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, 68. American Mathematical Society, Providence, RI (1999)

    Book  Google Scholar 

  16. Dijkgraaf, R., Verlinde, H., Verlinde, E.: Topological strings in \(d<1\). Nuclear Phys. B 352(1), 59–86 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  17. Freiermuth, H., Trautmann, G.: On the moduli scheme of stable sheaves supported on cubic space curves. Am. J. Math. 126(2), 363–393 (2004)

    Article  MathSciNet  Google Scholar 

  18. Gopakumar, R., Vafa, C.: M-theory and topological strings II. arXiv:hep-th/9812127

  19. Horja, R.P.: Derived category automorphisms from mirror symmetry. Duke Math. J. 127, 1–34 (2005)

    Article  MathSciNet  Google Scholar 

  20. Hosono, S., Saito, M., Takahashi, A.: Relative Lefschetz actions and BPS state counting. Int. Math. Res. Not. 15, 783–816 (2001)

    Article  MathSciNet  Google Scholar 

  21. Ionel, E.N., Parker, T.: The Gopakumar–Vafa formula for symplectic manifolds. Ann. Math. (2) 187(1), 1–64 (2018)

    Article  MathSciNet  Google Scholar 

  22. Katz, S.: Genus zero Gopakumar–Vafa invariants of contractible curves. J. Differ. Geom. 79, 185–195 (2008)

    Article  MathSciNet  Google Scholar 

  23. Kiem, Y.H., Li, J.: Categorification of Donaldson–Thomas invariants via perverse sheaves. arXiv:1212.6444

  24. Klemm, A., Pandharipande, R.: Enumerative geometry of Calabi–Yau 4-folds. Commun. Math. Phys. 281(3), 621–653 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  25. Maulik, D., Toda, Y.: Gopakumar–Vafa invariants via vanishing cycles. Invent. Math. 213(3), 1017–1097 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  26. Monavari, S.: Private discussions

  27. Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. Invent. Math. 178, 407–447 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  28. Pantev, T., Toën, B., Vaquie, M., Vezzosi, G.: Shifted symplectic structures. Publ. Math. IHES 117, 271–328 (2013)

    Article  MathSciNet  Google Scholar 

  29. Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nuclear Phys. B 340(2–3), 281–332 (1990)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are grateful to Martijn Kool and Sergej Monavari for helpful discussions and warmly thank Sergej Monavari for his help in doing a computation using his program. This work is partially supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan. Y. C. is partially supported by JSPS KAKENHI Grant Number JP19K23397 and Newton International Fellowships Alumni 2019. Y. T. is partially supported by Grant-in Aid for Scientific Research grant (No. 26287002) from MEXT, Japan.

Author information

Authors and Affiliations

  1. Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Yalong Cao & Yukinobu Toda

Authors
  1. Yalong Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yukinobu Toda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yalong Cao.

Additional information

Communicated by H. T. Yau

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

Cao, Y., Toda, Y. Gopakumar–Vafa Type Invariants on Calabi–Yau 4-Folds via Descendent Insertions. Commun. Math. Phys. 383, 281–310 (2021). https://doi.org/10.1007/s00220-020-03897-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-020-03897-9

Search

Navigation