Skip to main content
SpringerLink
Log in

Kähler Reduction of Metrics with Holonomy G2

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

Abstract

A torsion-free G 2 structure admitting an infinitesimal isometry such that the quotient is a Kähler manifold is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G 2.

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. Abbena, E., Garbiero, S., Salamon, S.: Almost Hermitian geometry of 6-dimensional nilmanifolds. Ann. Scuola Norm. Sup. Pisa (4) 30, 147–170 (2001)

    Google Scholar 

  2. Abbena, E., Garbiero, S., Salamon, S.: Hermitian geometry on the Iwasawa manifold. Boll. Un. Mat. Ital. 11-B, 231–249 (1997)

    Google Scholar 

  3. Acharya, B., Witten, E.: Chiral fermions from manifolds of G 2 holonomy. Available at arXiv: hep-th/0109152

  4. Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Hamiltonian 2-forms in Kähler geometry, I. Available at arXiv:math.DG/0202280

  5. Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: The geometry of weakly self-dual Kähler surfaces. Compositio Math. 135, 279–322 (2003)

    Article  MATH  Google Scholar 

  6. Apostolov, V., Gauduchon, P.: The Riemannian Goldberg-Sachs Theorem. Internat. J. Math. 8, 421–439 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Armstrong, J.: An ansatz for Almost-Kähler, Einstein 4-manifolds. J. reine angew. Math. 542, 53–84 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Atiyah, M., Witten, E.: M-theory dynamics on a manifold of G 2 holonomy. Adv. Theor. Math. Phys. 6, 1–106 (2003)

    Google Scholar 

  9. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1984

  10. Bedford, E., Kalka, M.: Foliations and the complex Monge-Ampère equation. Commun. Pure Appl. Math. 30, 543–571 (1977)

    MATH  Google Scholar 

  11. Behrndt, K., Dall’Agata, G., Lüst, D., Mahapatra, S.: Intersecting 6-branes from new 7-manifolds with G 2-holonomy. JHEP 8, no. (27) (2002), 24 pp.

  12. Brandhuber, A., Gomis, J., Gubser, S.S., Gukov, S.: Gauge theory at large N and new G 2 holonomy metrics. Nucl. Phys. B 611, 179–204 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Bryant, R.L., Salamon, S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829–850 (1989)

    MathSciNet  MATH  Google Scholar 

  15. Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G2 structures. In: Differential Geometry, Valencia 2001. River Edge, NJ: World Sci. Publishing, 2002, pp. 115–133

  16. Cvetic, M., Gibbons, G.W., Lü, H., Pope, C.N.: Almost special holonomy in type IIA and M-theory. Nucl. Phys. B 638, 186–206 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Derdziński, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math. 49, 405–433 (1983)

    MathSciNet  Google Scholar 

  18. Derdziński, A., Maschler, G.: Local classification of conformally-Einstein Kähler metrics in higher dimensions. Proc. London Math. Soc. (3) 87, 779–819 (2003)

  19. Fernández, M., Gray, A.: Riemannian manifolds with structure group G 2. Ann. Mat. Pura Appl. 32, 19–45 (1982)

    MathSciNet  Google Scholar 

  20. Gibbons, G.W., Lü, H., Pope, C.N., Stelle, K.S.: Supersymmetric domain walls from metrics of special holonomy. Nucl. Phys. B 623, 3–46 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hitchin, N.J.: Stable forms and special metrics. In: Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Contemp. Math. 288, Providence RI: Am. Math. Soc., 2001, pp. 70–89

  22. Hitchin, N.J.: The geometry of three-forms in six dimensions. J. Diff. Geom. 55, 547–576 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55, 59–126 (1989)

    MATH  Google Scholar 

  24. Joyce, D.D.: U(1)-invariant special Lagrangian 3-folds in ℂ3 and special Lagrangian fibrations. Turkish J. Math. 27, 99–114 (2003)

    MATH  Google Scholar 

  25. Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford: Oxford University Press, 2000

  26. Ketsetzis, G., Salamon, S.: Complex structures on the Iwasawa manifold. Adv. Geom. 4, 165–179 (2004)

    Google Scholar 

  27. LeBrun, C.: Einstein metrics on complex surfaces. In: Geometry and Physics (Aarhus, 1995), Lect. Notes Pure Appl. Math. Vol. 184, New York: Dekker, 1997, pp. 167–176

  28. Nurowski, P., Przanowski, M.: A four-dimensional example of Ricci flat metric admitting almost Kähler non-Kähler structure. Classical Quant. Grav. 16, L9–L13 (1999)

    Google Scholar 

  29. Salamon, S.: Riemannian geometry and holonomy groups. Pitman Research Notes in Mathematics Vol. 201, London-New York: Longman, 1989

  30. Santillan, O.P.: A construction of G 2 holonomy spaces with torus symmetry. Nucl. Phys. B 660, 169–193 (2003)

    Article  MATH  Google Scholar 

  31. Sekigawa, K.: On some compact Einstein almost-Kähler manifolds. J. Math. Soc. Japan 36, 677–684 (1987)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, UQAM, C.P. 8888, Succ. Centre-ville, Montréal (Québec), H3C 3P8, Canada

    Vestislav Apostolov

  2. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Simon Salamon

Authors
  1. Vestislav Apostolov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon Salamon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vestislav Apostolov.

Additional information

Communicated by G.W. Gibbons

Rights and permissions

Reprints and permissions

About this article

Cite this article

Apostolov, V., Salamon, S. Kähler Reduction of Metrics with Holonomy G2 . Commun. Math. Phys. 246, 43–61 (2004). https://doi.org/10.1007/s00220-003-1014-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-1014-2

Keywords

Search

Navigation