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Darboux–Halphen–Ramanujan Vector Field on a Moduli of Calabi-Yau Manifolds

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Abstract

In this paper we obtain an ordinary differential equation \(\mathsf{H}\) from a Picard–Fuchs equation associated with a nowhere vanishing holomorphic \(n\)-form. We work on a moduli space \(\mathsf{T }\) constructed from a Calabi–Yau \(n\)-fold \(W\) together with a basis of the middle complex de Rham cohomology of \(W\). We verify the existence of a unique vector field \(\mathsf{H}\) on \(\mathsf{T }\) such that its composition with the Gauss–Manin connection satisfies certain properties. The ordinary differential equation given by \(\mathsf{H}\) is a generalization of differential equations introduced by Darboux, Halphen and Ramanujan.

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References

  1. Batyrev, V.V., Van Starten, D.: Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric variety. Commun. Math. Phys. 168, 169–178 (1995)

    Article  Google Scholar 

  2. Bogner, M.: On Differential Operators of Calabi-Yau Type, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-University of Minz (2012)

  3. Brioschi, Fr: Sur un systéme d’équations différetielles. Comptes Rendus des sánces de l’Acad. des Sci. 92, 1389–1393 (1881)

    MATH  Google Scholar 

  4. Calabi, E.: The space of Kähler metrics. Proc. Int. Congr. Math., Amesterdam 1954(2), 206–207 (1956)

    Google Scholar 

  5. Calabi, E.: On Kähler manifolds with vanishing canonical class. In: algebraic geometry and topology, a simposium in honer of S, pp. 78–89. Lefschetz, Princeton University Press(1957)

  6. Candelas, P., De La Ossa, X.C., Green, P.S., Parkes, L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Phys. B 359(1), 21–74 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nuclear Phys. 258, 47–74 (1985)

    Article  MathSciNet  Google Scholar 

  8. Chen, Y-H., Yang, Y., Yui, N.: Monodromy of Picard–Fuchs differential equations for Calabi–Yau threefolds. J. Reine Angew. Math. 616, 167–203 (2008)

  9. Darboux, G.: Sur la théorie des coordonnées curvilignes et les systémes orthogonaux. Ann. Ec. Normale Supérieure 7, 101–150 (1878)

    MATH  MathSciNet  Google Scholar 

  10. Doran, C.F., Morgan, J.W.: Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi–Yau threefolds, Mirror symmetry. V, AMS/IP Stud. Adv. Math. 38, 517–537 (2006)

    MathSciNet  Google Scholar 

  11. Greene, B.R., Plesser, M.R.: Duality in Calabi–Yau moduli space. Nuclear Phys. 338(1), 15–37 (1990)

    Article  MathSciNet  Google Scholar 

  12. Gross, M.W., Huybrechts, D., Joyce, D.D.: Calabi–Yau manifolds and related geometries, lectures at a summer school in Nordfjordeid, Norway, June 2001, Springer (2003)

  13. Guest, M.A.: From Quantum Cohomology to Integrable Systems (Oxford Graduate Texts in Mathematics). Oxford University Press Inc., New York (2008)

  14. Guillot, A.: Champs quadratiques uniformisables. These de doctorat, Ecole Normale Suprieure de Lyon (2001)

  15. Guillot, Adolfo: Semicomplete meromorphic vector fields on complex surfaces. J. Reine Angew. Math. 667, 27–65 (2012)

    MATH  MathSciNet  Google Scholar 

  16. Halphen, G.H.: Sur des fonctions qui proviennent de l’équation de Gauss. C. R. Acad. Sci. Paris 92, 856–859 (1881)

    MATH  Google Scholar 

  17. Halphen, G.H.: Sur un systéme d’équations différetielles. C. R. Acad. Sci. Paris 92, 1101–1103 (1881)

    MATH  Google Scholar 

  18. Halphen, G.H.: Sur certains systéme d’équations différetielles. C. R. Acad. Sci. Paris 92, 1404–1407 (1881)

    MATH  Google Scholar 

  19. Klemm, A., Theisen, S.: Mirror maps and instanton sums for complete intersections in weighted projective space. Modern Phys. Lett. A 9(20), 1807–1817 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Libgober, A., Teitelbaum, J.: Lines on Calabi–Yau complete intersections, mirror symmetry, and Picard–Fuchs equations. Int. Math. Res. Notices 1, 29–39 (1993)

    Article  MathSciNet  Google Scholar 

  21. Morrison, David R.: Picard–Fuchs Equations and Mirror Maps for Hypersurfaces, Essays on Mirror Manifolds. Int. Press, Hong Kong (1992)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Movasati, H.: Multiple Integrals and Modular Differential Equations, 28th Brazilian Mathematics Colloquium. Instituto de Matemática Pura e Aplicada, IMPA (2011)

  24. Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916)

    Google Scholar 

  25. Van der Put, M., Singer, M.F.: Galois theory of linear differential equations, vol. 328 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, (2003)

  26. Voisin, C.: Cambridge studies in advanced mathematics, vol. 76. Hodge Theory and Complex Algebraic Geometry. I. Cambridge University Press, Cambridge (2002). (Translated from the French original by Leila Schneps)

    Chapter  Google Scholar 

  27. Yau, S.T.: On the Ricci curvature of a compct Kähler manifold and the complex Monge–Ampére equations I. Commun. Pure Appl. Math. 31, 339–411 (1978)

    Article  MATH  Google Scholar 

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Acknowledgments

Here I would like to express my very great appreciation to Hossein Movasati, my Ph.D. supervisor, who always was available and I used his valuable and constructive suggestions and helps during the planning and development of this work. I wish to thank IMPA for preparing such an excellent academic environment. This work has been done during my Ph.D. and I am grateful to have economic supports of “CNPq-TWAS Fellowships Programme” during this period.

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

    Younes Nikdelan

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  1. Younes Nikdelan
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Nikdelan, Y. Darboux–Halphen–Ramanujan Vector Field on a Moduli of Calabi-Yau Manifolds. Qual. Theory Dyn. Syst. 14, 71–100 (2015). https://doi.org/10.1007/s12346-014-0129-5

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