Skip to main content
SpringerLink
Log in

Holomorphic Anomaly Equations for the Formal Quintic

  • Original Article
  • Published:
Peking Mathematical Journal Aims and scope Submit manuscript

Abstract

We define a formal Gromov–Witten theory of the quintic threefold via localization on \({\mathbb {P}}^4\). Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov–Witten theory of the quintic threefold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.

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. For stability, marked points are required in genus 0 and positive degree is required in genus 1.

  2. A second proof (in most cases) can be found in [10].

  3. The negative exponent denotes the dual: \(\mathsf {S}\) is a line bundle and \(\mathsf {S}^{-5}=(\mathsf {S}^\star )^{\otimes 5}\).

  4. Since the formal quintic theory is homogeneous of degree 0, the specialization \(\lambda _i = \zeta ^i\lambda _0\) could also be taken as in [22]. However, for other specializations, we expect the finite generation of Theorem 1 and the holomorphic anomaly equation of Theorem 2 to take different forms. A study of the dependence on specialization will appear in [21]. For local \(\mathbb {P}^2\) considered in [22] and \(\mathbb {C}^3/\mathbb {Z}_3\) considered in [23], the theories are independent of specialization.

  5. Our functions \(K_2\) and \(A_{2k}\) are normalized differently with respect to \(C_0\) and \(C_1\). The dictionary to exactly match the notation of [1, (2.52)] is to multiply our \(K_2\) by \((C_0C_1)^2\) and our \(A_{2k}\) by \((C_0C_1)^{2k}\).

  6. See Remark 27.

  7. Corresponding to a stratum of the moduli space of stable curves \(\overline{M}_{g,n}\).

  8. Self-edges correspond to loops of \({\mathsf {T}}\)-invariant rational curves.

  9. The moduli spaces \(\overline{Q}_{0,0}({\mathbb {P}}^m,d)\) and \(\overline{Q}_{0,1}({\mathbb {P}}^m,d)\) are empty by the definition of a stable quotient.

  10. The associated weights on \(H^0({\mathbb {P}}^4,\mathcal {O}_{{\mathbb {P}}^4}(1))\) are \(\lambda _0,\lambda _1,\lambda _2,\lambda _3,\lambda _4\) and so match the conventions of Sect. 1.2.

  11. Equation (12) is the definition of \(e(\text {Obs})\). The right side of (12) is defined after localization as explained in Sect. 1.2.

  12. See Sections 2 and 5 of [6].

  13. In Gromov–Witten theory, a parallel relation is obtained immediately from the WDDV equation and the string equation. Since the map forgetting a point is not always well-defined for quasimaps, a different argument is needed here [8].

  14. We follow here the notation of [26] for \(B_k\).

  15. We use the variables \(x_1\) and \(x_2\) here instead of x and y.

  16. We follow here the notation of Sect. 2.

  17. Flags are either half-edges or markings.

  18. In case e is self-edge, \(v_1=v_2\).

References

  1. Alim, M., Scheidegger, E., Yau, S.-T., Zhou, J.: Special polynomial rings, quasi modular forms and duality of topological strings. Adv. Theor. Math. Phys. 18, 401–467 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128, 45–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279–304 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ciocan-Fontanine, I., Kim, B.: Moduli stacks of stable toric quasimaps. Adv. Math. 225, 3022–3051 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ciocan-Fontanine, I., Kim, B.: Wall-crossing in genus zero quasimap theory and mirror maps. Algebr. Geom. 1, 400–448 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciocan-Fontanine, I., Kim, B.: Big I-functions. In: Development of Moduli Theory—Kyoto, vol. 69. Advanced Studies in Pure Mathematics. Mathematical Society of Japan, Tokyo, pp. 323–347 (2016)

  7. Ciocan-Fontanine, I., Kim, B.: Quasimap Wallcrossings and Mirror Symmetry. arXiv:1611.05023

  8. Ciocan-Fontanine, I., Kim, B.: Higher genus quasimap wall-crossing for semi-positive targets. JEMS 19, 2051–2102 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ciocan-Fontanine, I., Kim, B., Maulik, D.: Stable quasimaps to GIT quotients. J. Geom. Phys. 75, 17–47 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clader, E., Janda, F., Ruan, Y.: Higher Genus Quasimap Wall-Crossing Via Localization. arXiv:1702.03427

  11. Cooper, Y., Zinger, A.: Mirror symmetry for stable quotients invariants. Michigan Math. J. 63, 571–621 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  13. Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Algebraic Geometry—Santa Cruz 1995, Proceedings on Symposium of Pure Mathematics, vol. 62, Part 2. American Mathematical Society, Providence, RI, pp. 45–96 (1997)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Givental, A.: Elliptic Gromov–Witten invariants and the generalized mirror conjecture. In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997). World Scientific, River Edge, pp. 107–155 (1998)

  16. Givental, A.: Semisimple Frobenius structures at higher genus. Int. Math. Res. Not. 23, 613–663 (2001)

    MATH  Google Scholar 

  17. Graber, T., Pandharipande, R.: Localization of virtual classes. Invent. Math. 135, 487–518 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, B., Lho, H.: Mirror theorem for elliptic quasimap invariants. Geom. Top. 22, 1459–1481 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kontsevich, M.: Enumeration of rational curves via torus actions. In: The Moduli Space of Curves (Texel Island, 1994). Progress in Mathematics, vol. 129. Birkhäuser, Boston, pp. 335–368 (1995)

  20. Lee, Y.-P., Pandharipande, R.: Frobenius Manifolds, Gromov–Witten Theory and Virasoro Constraints (2004). https://people.math.ethz.ch/~rahul/

  21. Lho, H.: Equivariant Holomorphic Anomaly Equation for the Formal Quintic (in preparation)

  22. Lho, H., Pandharipande, R.: Stable quotients and the holomorphic anomaly equation. Adv. Math. 332, 349–402 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lho, H., Pandharipande, R.: Crepant Resolution and the Holomorphic Anomaly Equation for \({\mathbb{C}}^{3}/{\mathbb{Z}}_{3}\). arXiv:1804.03168

  24. Marian, A., Oprea, D., Pandharipande, R.: The moduli space of stable quotients. Geom. Topol. 15, 1651–1706 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pandharipande, R., Pixton, A., Zvonkine, D.: Relations on \(\overline{M}_{g, n}\) via 3-spin structures. JAMS 28, 279–309 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Yamaguchi, S., Yau, S.-T.: Topological string partition functions as polynomials. J. High Energy Phys. 2004(7) (2004)

  27. Zagier, D., Zinger, A.: Some properties of hypergeometric series associated with mirror symmetry in modular forms and string duality. Fields Inst. Commun. 54, 163–177 (2008)

    MATH  Google Scholar 

Download references

Acknowledgements

We thank I. Ciocan-Fontanine, E. Clader, Y. Cooper, B. Kim, A. Klemm, Y.-P. Lee, A. Marian, M. Mariño, D. Maulik, D. Oprea, E. Scheidegger, Y. Toda, and A. Zinger for discussions over the years about the moduli space of stable quotients and the invariants of Calabi–Yau geometries. The work of Q. Chen, F. Janda, S. Guo, and Y. Ruan as presented at the Workshop on higher genus is crucial for the wider study of the formal (and true) quintic. We are very grateful to them for sharing their ideas with us. R. P. was partially supported by SNF-200020182181, ERC-2012-AdG-320368-MCSK, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. H. L. was supported by the Grants ERC-2012-AdG-320368-MCSK and ERC-2017-AdG-786580-MACI. This project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 786580).

Author information

Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Zurich, Switzerland

    Hyenho Lho & Rahul Pandharipande

Authors
  1. Hyenho Lho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rahul Pandharipande
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyenho Lho.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lho, H., Pandharipande, R. Holomorphic Anomaly Equations for the Formal Quintic. Peking Math J 2, 1–40 (2019). https://doi.org/10.1007/s42543-018-0008-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42543-018-0008-0

Keywords

Search

Navigation