Skip to main content
SpringerLink
Log in

Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Sasakian manifolds are odd-dimensional counterpart to Kähler manifolds. They can be defined as contact manifolds equipped with an invariant Kähler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kähler supersymmetry algebra is associated to a Kähler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.

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. Here, as elsewhere, “differential operators on the de Rham algebra” are understood in the algebraic sense as above.

  2. The assumption (*) and (**) hold for Sasakian manifolds (Proposition 3.3 and Remark 3.4) and Vaisman manifolds (Remark 7.7).

  3. The statement of Theorem 6.6 is global, however, the proof of Theorem 6.6, step 1 is local, and this statement is essentially identical to (8.7).

References

  1. Álvarez López, J.A.: A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois. J. Math. 33, 79–92 (1989)

    MathSciNet  MATH  Google Scholar 

  2. Álvarez López, J.A.: Duality in the spectral sequence of Riemannian foliations. Am. J. Math. 111(6), 905–926 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belgun, F.A.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317, 1–40 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Biswas, I., Fernández, M., Muñoz, V.: Tralle, Aleksy On formality of Sasakian manifolds. J. Topol. 9(1), 161–180 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Progress in Mathematics, 203. Birkhäuser Boston, Inc., Boston (2010)

  6. Boyer, C., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs, Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  7. Brylinsky, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Birkhäuser, Oxford (2008)

    Google Scholar 

  8. Cappelleti-Montano, B., De Nicola, A., Marrero, J.C., Yudin, I.: Sasakian nilmanifolds. Int. Math. Res. Not. IMRN 72, 6648–6660 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Coutinho, S.C.: A Primer of Algebraic D-modules. London Mathematical Society Study, vol. 33. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  10. Domínguez, D.: A finiteness theorem for transitive foliations and flat vector bundles Israel. J. Math. 107, 253–287 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Dragomir, S., Ornea, L.: Locally Conformally Kähler Manifolds, Progress in Mathematics 55, Birkhäuser, (1998)

  12. El Kacimi-Alaoui, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compositio Math. 73(1), 57–106 (1990)

    MathSciNet  MATH  Google Scholar 

  13. El Kacimi-Alaoui, A., Sergiescu, V., Hector, G.: La cohomologie basique d’un feuilletage riemannien est de dimension finie. Math. Z. 188(4), 593–599 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Figueroa-O’Farrill, J.M., Koehl, C., Spence, B.: Supersymmetry and the cohomology of (hyper)Kähler manifolds. Nuclear Phys. B 503(3), 614–626 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fedoruk, S., Ivanov, E., Smilga, A.: Generic HKT geometries in the harmonic superspace approach. J. Math. Phys. 59(8), 083501 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fedoruk, S., Smilga, A.: Bi-HKT and bi-Kähler supersymmetric sigma models. J. Math. Phys. 57(4), 042103 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  18. Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry. Wiley-Intersience, New York (1978)

    MATH  Google Scholar 

  19. Hattori, A.: Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I(8), 289–331 (1960)

    MathSciNet  MATH  Google Scholar 

  20. Huang, T.: \(L^2\) harmonic forms on complete special holonomy manifolds. Ann. Global Anal. Geom. 56(1), 17–36 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Istrati, N., Otiman, A., Pontecorvo, M.: On a class of Kato manifolds. Int. Math. Res. Not. IMRN 7, 5366–5412 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kamishima, Y., Ornea, L.: Geometric flow on compact locally conformally Kähler manifolds. Tohoku Math. J. 57(2), 201–221 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kashiwada, T., Sato, S.: On harmonic forms in compact locally conformal Kähler manifolds with the parallel Lee form. Ann. Fac. Sci. Univ. Nat. Zaïre (Kinshasa) Sect. Math. Phys. 6(1–2), 17–29 (1980)

    MathSciNet  MATH  Google Scholar 

  24. Kim, H., Saberi, I.: Real homotopy theory and supersymmetric quantum mechanics. J. Math. Phys. 59(7), 072102 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nagy, P.-A.: Un principe de séparation des variables pour le spectre du laplacien des formes différentielles et applications, Ph. D. Thesis, Université de Savoie (2001)

  26. Ornea, L., Verbitsky, M.: Structure theorem for compact Vaisman manifolds. Math. Res. Lett. 10, 799–805 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ornea, L., Verbitsky, M.: LCK rank of locally conformally Kähler manifolds with potential. J. Geom. Phys. 107, 92–98 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Schmude, J.: Laplace operators on Sasaki-Einstein manifolds. JHEP 1404, 008 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schwarz, G.W.: On the de Rham cohomology of the leaf space of a foliation. Topology 13, 185–187 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sergiescu, V.: Quelques problèmes concernant les feuilletages et les groupes de difféomorphismes, Université de Lille 1, Thèse de doctorat en Mathématiques pures, (1986), http://ori.univ-lille1.fr/notice/view/univ-lille1-ori-39175

  31. Sergiescu, V.: Sur la suite spectrale d’un feuilletage riemannien Proceedings of the XIXth National Congress of the Mexican Mathematical Society, Vol. 2 (Guadalajara, 1986), pp. 33–39, Aportaciones Mat. Comun., 4, Soc. Mat. Mexicana, México (1987)

  32. Smilga, A.V.: Supercharges in the hyper-Kähler with torsion supersymmetric sigma models. J. Math. Phys. 53(12), 122105 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tachibana, S.: On harmonic tensors in compact Sasakian spaces. Tohoku Math. J. 17, 271–284 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tievsky, A.: Analogues of Kähler Geometry on Sasakian Manifolds, Ph.D. Thesis, M.I.T., (2008)

  35. Tsukada, K.: The canonical foliation of a compact generalized Hopf manifold. Differ. Geom. Appl. 11(1), 13–28 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vaisman, I.: Generalized Hopf manifolds. Geom. Dedicata. 13, 231–255 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  37. Verbitsky, M.: Vanishing theorems for locally conformal hyperkähler manifolds. Proc. Steklov Inst. 246, 54–79 (2004). arxiv:math.DG/0302219

    Google Scholar 

  38. Verbitsky, M.: Hyperkähler manifolds with torsion, supersymmetry and Hodge theory. Asian J. Math. 6(4), 679–712 (2002). arxiv:math.AG/0112215

    Article  MathSciNet  MATH  Google Scholar 

  39. Verbitsky, M.: Manifolds with parallel differential forms and Kaehler identities for \(G_2\)-manifolds. J. Geom. Phys. 61(6), 1001–1016 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Verbitsky, M.: Hodge theory on nearly Kähler manifolds Geom. Topol. 15(4), 2111–2133 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Verbitsky, M., Vuletescu, V., Ornea, L.: Classification of non-Kähler surfaces and locally conformally Kähler geometry. Russian Math. Surv. 76, 261–290 (2021)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

L.O. thanks IMPA (Rio de Janeiro) and HSE (Moscow) for financial support and excellent research environment during the preparation of this paper. Both authors thank P.-A. Nagy for useful discussions. We are grateful to Richard Eager for the reference to [28] and to Nikita Klemyatin for pointing out some errors in a first version of the paper. Many thanks to the referee and the editor for very useful comments.

Author information

Authors and Affiliations

  1. University of Bucharest, Faculty of Mathematics and Informatics, 14 Academiei str., 70109, Bucharest, Romania

    Liviu Ornea

  2. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Str., 010702, Bucharest, Romania

    Liviu Ornea

  3. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, 110 Jardim Botânico, CEP 22460-320, Rio de Janeiro, RJ, Brasil

    Misha Verbitsky

  4. Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Str., Moscow, Russia

    Misha Verbitsky

Authors
  1. Liviu Ornea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Misha Verbitsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liviu Ornea.

Ethics declarations

Data availability statement

Not applicable.

Additional information

Publisher's Note

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

Liviu Ornea is partially supported by a grant of Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0065, within PNCDI III.

Misha Verbitsky is partially supported by by the HSE University Basic Research Program, FAPERJ E-26/202.912/2018 and CNPq - Process 313608/2017-2.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ornea, L., Verbitsky, M. Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds. manuscripta math. 170, 629–658 (2023). https://doi.org/10.1007/s00229-021-01358-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-021-01358-8

Keywords

Mathematics Subject Classification

Search

Navigation