Skip to main content
SpringerLink
Log in

Gauss–Manin Connection in Disguise: Calabi–Yau Threefolds

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

Abstract

We describe a Lie Algebra on the moduli space of non-rigid compact Calabi–Yau threefolds enhanced with differential forms and its relation to the Bershadsky–Cecotti–Ooguri–Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions \({{\bf F}_{g}^{\rm alg}, g \geq 1}\), which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck’s algebraic de Rham cohomology and on the algebraic Gauss–Manin connection. In this way, we recover a result of Yamaguchi–Yau and Alim–Länge in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi–Yau threefolds correspond to coordinates on such a moduli space. We discuss the mirror quintic as an example.

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

References

  1. Alim M., Lange J.D.: Polynomial Structure of the (Open) Topological String Partition Function. JHEP 0710, 045 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  2. Alim M.: Lectures on Mirror Symmetry and Topological String Theory. Open Probl. Surv. Contemp Math. 6, 1 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alim M., Lange J.D., Mayr P.: Global Properties of Topological String Amplitudes and Orbifold Invariants. JHEP 1003, 113 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. 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(2), 401–467 (2014)

  5. Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Holomorphic Anomalies in Topological Field theories. Nucl. Phys. B 405, 279–304 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311–428 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Ceresole, A., D’Auria, R., Ferrara, S., Lerche, W., Louis, J., Regge, T.: Picard–Fuchs equations, special geometry and target space duality. In: Mirror symmetry, II, vol. 1, AMS/IP Stud. Adv. Math., pp. 281–353. American Mathematical Society, Providence (1997)

  8. Candelas P., de la Ossa X.: Moduli Space of Calabi–Yau Manifolds. Nucl. Phys. B 355, 455–481 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Candelas P., DeLaOssa X.C., Green P.S., Parkes L.: A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21–74 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Cox, D.A., Katz, S.: Mirror symmetry and algebraic geometry, vol. 68, Mathematical Surveys and Monographs. American Mathematical Society, Providence (1999)

  11. Costello, K.J., Li, S.: Quantum BCOV theory on Calabi–Yau manifolds and the higher genus B-model (2012)

  12. Deligne, P.: Private letter. http://w3.impa.br/~hossein/myarticles/deligne6-12-2008 (2009)

  13. Deligne, P., Milne, J.S., Ogus, A., Shih, K.-Y.: Hodge cycles, motives, and Shimura varieties, vol. 900, Lecture Notes in Mathematics. Springer-Verlag, Berlin (1982)

  14. Givental, A.B.: Equivariant Gromov–Witten invariants. Int. Math. Res. Notices (13):613–663 (1996)

  15. Grothendieck A.: On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 29, 95–103 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guillot A.: Sur les équations d’Halphen et les actions de \({{\rm sl}_{2}({\bf c})}\). Publ. Math. Inst. Hautes Études Sci. 105, 221–294 (2007)

    Article  MathSciNet  Google Scholar 

  17. Huang M.-X., Klemm A., Quackenbush S.: Topological String Theory on Compact Calabi–Yau: Modularity and Boundary Conditions. Lect. Notes Phys. 757, 45–102 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  18. Hosono, S.: BCOV ring and holomorphic anomaly equation. In: New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008), Advanced Studies in Pure Mathematics, vol. 59, pp. 79–110. A publication of the Mathematical Society of Japan, Tokyo (2010)

  19. Katz N.M., Oda T.: On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. 8, 199–213 (1968)

    MathSciNet  MATH  Google Scholar 

  20. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, (1994)) pp. 120–139. Birkhäuser, Basel (1995)

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Morrison, D.R.: Picard–Fuchs equations and mirror maps for hypersurfaces. In: Essays on mirror manifolds, pp. 241–264. International Press, Hong Kong (1992)

  23. Morrison, D.R.: Mathematical aspects of mirror symmetry. In Complex algebraic geometry (Park City, UT, 1993), vol. 3 of IAS/Park City Math. Ser., pp. 265–327. American Mathematical Society, Providence (1997)

  24. Movasati, H.: On elliptic modular foliations. Indag. Math. (N.S.) 19(2), 263–286 (2008) arXiv:0806.3926v1 [math.AG]

  25. Movasati H.: Eisenstein type series for Calabi–Yau varieties. Nuclear Phys. B 847(2), 460–484 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Movasati H.: Modular-type functions attached to mirror quintic Calabi–Yau varieties. Math. Zeit. 281(3), 907–929 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Movasati H.: On Ramanujan relations between Eisenstein series. Manuscr. Math. 139(3–4), 495–514 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Movasati H.: Quasi-modular forms attached to elliptic curves, i. Ann. Math. Blaise Pascal 19(2), 307–377 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Movasati, H.: Quasi-modular forms attached to Hodge structures. In: Arithmetic and geometry of K3 surfaces and Calabi–Yau threefolds, vol 67 of Fields Inst. Commun., pp. 567–587. Springer, New York (2013)

  30. Nikdelan Y.: Darboux–Halphen–Ramanujan vector field on a moduli of Calabi–Yau manifolds. Qual. Theory Dyn. Syst. 14(1), 71100 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Strominger A.: Special geometry. Commun. Math. Phys. 133, 163–180 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Todorov, A.: Local and global theory of the moduli of polarized Calabi–Yau manifolds. In: Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), vol. 19, pp. 687–730 (2003)

  33. Viehweg, E.: Quasi-projective moduli for polarized manifolds, volume 30 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin (1995)

  34. Voisin, C.: Mirror symmetry, vol. 1, SMF/AMS Texts and Monographs. American Mathematical Society, Providence; Société Mathématique de France, Paris (1999) (Translated from the 1996 French original by Roger Cooke)

  35. Witten E.: Topological Sigma Models. Commun. Math. Phys. 118, 411 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Witten, E.: Mirror manifolds and topological field theory. In: Essays on mirror manifolds, pp. 120–158. International Press, Hong Kong (1992)

  37. Witten, E.: Quantum background independence in string theory (1993) arXiv:hep-th/9306122

  38. Yamaguchi S., Yau S.-T.: Topological string partition functions as polynomials. JHEP 07, 047 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  39. Zinger A.: The reduced genus 1 Gromov–Witten invariants of Calabi–Yau hypersurfaces. J. Am. Math. Soc. 22(3), 691–737 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Murad Alim & Shing-Tung Yau

  2. Jefferson Physical Laboratory, 17 Oxford Street, Cambridge, MA, 02138, USA

    Murad Alim

  3. Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil

    Hossein Movasati

  4. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstrasse 1, D-79104, Freiburg, Germany

    Emanuel Scheidegger

Authors
  1. Murad Alim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hossein Movasati
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emanuel Scheidegger
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shing-Tung Yau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hossein Movasati.

Additional information

Communicated by H. T. Yau

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alim, M., Movasati, H., Scheidegger, E. et al. Gauss–Manin Connection in Disguise: Calabi–Yau Threefolds. Commun. Math. Phys. 344, 889–914 (2016). https://doi.org/10.1007/s00220-016-2640-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-016-2640-9

Keywords

Search

Navigation