Skip to main content
SpringerLink
Log in

Universal Maslov class of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We show that in the case of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein symplectic manifold, a certain universal 1-cohomology class, analogous to the Maslov class, can be defined. In contrast to the Maslov index, the presented class is directly related to the minimality problem for Lagrangian submanifolds if the ambient pseudo-Einstein manifold admits a Kähler-Einstein metric. We interpret the presented class geometrically as a certain obstruction to the continuation of one-dimensional supercycles from the Lagrangian submanifold to the ambient symplectic manifold.

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. V. I. Arnold and A. G. Givental, Symplectic Geometry [in Russian], Udmurtia State Univ., Izhevsk (2000).

    Google Scholar 

  2. M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization [in Russian], Nauka, Moscow (1991); English transl., Amer. Math. Soc., Providence, R. I. (1993).

    MATH  Google Scholar 

  3. K. Fukaya, “Morse homotopy, A ∞-category, and Floer homologies,” MSRI Preprint No. 020-94, Math. Sci. Res. Inst., Berkeley, Calif. (1994); “Morse homotopy, A ∞-category, and Floer homologies,” in: Proc. GARC Workshop on Geometry and Topology ’93 (Lect. Notes Ser. Seoul, Vol. 18), Seoul National Univ., Seoul (1993), pp. 1–102.

    Google Scholar 

  4. N. A. Tyurin, “Maslov class of lagrangian embedding to Kahler manifold,” math.SG/0602330 (2006).

  5. S. Donaldson and P. Kronheimer, The Geometry of Four-Manifolds, Clarendon, Oxford (1990).

    MATH  Google Scholar 

  6. A. L. Gorodentsev and A. N. Tyurin, Izv. Math., 65, 437–467 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  7. N. A. Tyurin, Izv. Math., 66, 611–629 (2002).

    Article  MathSciNet  Google Scholar 

  8. Y.-G. Oh, Comm. Pure Appl. Math., 46, 949–993 (1993).

    MATH  MathSciNet  Google Scholar 

  9. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York (1978).

    MATH  Google Scholar 

  10. M. Karasev, Russ. J. Math. Phys., 3, 393–400 (1995).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia

    N. A. Tyurin

  2. Moscow State University of Railway Engineering (MIIT), Moscow, Russia

    N. A. Tyurin

Authors
  1. N. A. Tyurin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 325–337, February, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tyurin, N.A. Universal Maslov class of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold. Theor Math Phys 150, 278–287 (2007). https://doi.org/10.1007/s11232-007-0021-4

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-007-0021-4

Keywords

Search

Navigation