Skip to main content
Log in

Symplectic half-flat solvmanifolds

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Andriot D.: New supersymmetric flux vacua with intermediate SU(2)-structure. J. High Energy Phys. 0808, 096 (2008)

    Article  MathSciNet  Google Scholar 

  2. Andriot D., Goi E., Minasian R., Petrini M.: Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory. J. High Energy Phys. 1105, 028 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bock, C.: On low-dimensional solvmanifolds. arXiv:0903.2926v4 [math.DG]

  4. Conti D.: Half-flat nilmanifolds. Math. Ann. 350(1), 155–168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conti D., Fernández M.: Nilmanifolds with a calibrated G 2-structure. Differ. Geom. Appl. 29, 493–506 (2011)

    Article  MATH  Google Scholar 

  6. Conti D., Tomassini A.: Special symplectic six-manifolds. Q. J. Math. 58, 297–311 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cortés V., Leistner T., Schäfer L., Schulte-Hengesbach F.: Half-flat structures and special holonomy. Proc. London Math. Soc. (3) 102(1), 113–158 (2011)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  9. Fernández M., de León M., Saralegui M.: A six dimensional symplectic solvmanifold without Kähler structures. Osaka J. Math. 33, 19–35 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Fino, A., Ugarte, L.: On the geometry underlying supersymmetric flux vacua with intermediate SU(2) structure. Classical Quantum Gravity 28(7), 075004, 21 pp. (2011)

    Google Scholar 

  11. Freibert M., Schulte-Hengesbach F.: Half-flat structures on decomposable Lie groups. Transform. Groups 17(1), 123–141 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Freibert, M., Schulte-Hengesbach, F.: Half-flat structures on indecomposable Lie groups. Transform. Groups. doi:10.1007/s00031-012-9190-9

  13. Gorbatsevich V.V.: Symplectic structures and cohomologies on some solv-manifolds. Siberian Math. J. 44(2), 260–274 (2003)

    Article  MathSciNet  Google Scholar 

  14. Graña, M., Minasian, R., Petrini, M., Tomasiello, A.: A scan for new N = 1 vacua on twisted tori. J. High Energy Phys. 0705, 031 (2007)

    Google Scholar 

  15. Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148(3), 47–157 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Hitchin, N.: Stable Forms and Special Metrics. Global Differential Geometry: The Mathematical Legacy of Alfred Gray, pp. 70–89. American Mathematical Society, Providence, RI (2001)

  18. Macrì, M.: Cohomological properties of unimodular six dimensional solvable Lie algebras. arXiv:1111.5958v2 [math.DG]

  19. Mubarakzyanov G.M.: Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 35(4), 104–116 (1963)

    Google Scholar 

  20. Schulte-Hengesbach F.: Half-flat structures on products of three-dimensional Lie groups. J. Geom. Phys. 60(11), 1726–1740 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shabanskaya, A.: Classification of six dimensional solvable indecomposable lie algebras with a codimension one nilradical over \({\mathbb{R}}\). Ph.D.Thesis, University of Toledo, Ohio (2011)

  22. Tomassini A., Vezzoni L.: On symplectic half-flat manifolds. Manuscripta Math. 125(4), 515–530 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tralle A., Oprea J.: Symplectic manifolds with no Kähler structures. Lectures Notes in Mathematics, 1661. Springer, Berlin (1997)

    Google Scholar 

  24. Turkowski P.: Solvable Lie algebras of dimension six. J. Math. Phys. 31, 1344–1350 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yamada T.: A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold. Geom. Dedicata 112, 115–122 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to L. Ugarte.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fernández, M., Manero, V., Otal, A. et al. Symplectic half-flat solvmanifolds. Ann Glob Anal Geom 43, 367–383 (2013). https://doi.org/10.1007/s10455-012-9349-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-012-9349-6

Keywords

Mathematics Subject Classification (2000)

Navigation