Annals of Global Analysis and Geometry

, Volume 43, Issue 4, pp 367–383 | Cite as

Symplectic half-flat solvmanifolds

  • M. Fernández
  • V. Manero
  • A. Otal
  • L. UgarteEmail author


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.


Symplectic half-flat structures Solvable Lie algebras Supersymmetric equations of type IIA 

Mathematics Subject Classification (2000)

Primary 53C15 Secondary 22E25 53C80 17B30 53C38 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andriot D.: New supersymmetric flux vacua with intermediate SU(2)-structure. J. High Energy Phys. 0808, 096 (2008)MathSciNetCrossRefGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bock, C.: On low-dimensional solvmanifolds. arXiv:0903.2926v4 [math.DG]Google Scholar
  4. 4.
    Conti D.: Half-flat nilmanifolds. Math. Ann. 350(1), 155–168 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Conti D., Fernández M.: Nilmanifolds with a calibrated G 2-structure. Differ. Geom. Appl. 29, 493–506 (2011)zbMATHCrossRefGoogle Scholar
  6. 6.
    Conti D., Tomassini A.: Special symplectic six-manifolds. Q. J. Math. 58, 297–311 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 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)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fernández M., Gray A.: Riemannian manifolds with structure group G 2. Ann. Mat. Pura Appl. 32, 19–45 (1982)CrossRefGoogle Scholar
  9. 9.
    Fernández M., de León M., Saralegui M.: A six dimensional symplectic solvmanifold without Kähler structures. Osaka J. Math. 33, 19–35 (1996)MathSciNetzbMATHGoogle Scholar
  10. 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. 11.
    Freibert M., Schulte-Hengesbach F.: Half-flat structures on decomposable Lie groups. Transform. Groups 17(1), 123–141 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Freibert, M., Schulte-Hengesbach, F.: Half-flat structures on indecomposable Lie groups. Transform. Groups. doi: 10.1007/s00031-012-9190-9
  13. 13.
    Gorbatsevich V.V.: Symplectic structures and cohomologies on some solv-manifolds. Siberian Math. J. 44(2), 260–274 (2003)MathSciNetCrossRefGoogle Scholar
  14. 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. 15.
    Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148(3), 47–157 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hitchin N.: The geometry of three-forms in six dimensions. J. Differ. Geom. 55, 547–576 (2000)MathSciNetzbMATHGoogle Scholar
  17. 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)Google Scholar
  18. 18.
    Macrì, M.: Cohomological properties of unimodular six dimensional solvable Lie algebras. arXiv:1111.5958v2 [math.DG]Google Scholar
  19. 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. 20.
    Schulte-Hengesbach F.: Half-flat structures on products of three-dimensional Lie groups. J. Geom. Phys. 60(11), 1726–1740 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 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)Google Scholar
  22. 22.
    Tomassini A., Vezzoni L.: On symplectic half-flat manifolds. Manuscripta Math. 125(4), 515–530 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Tralle A., Oprea J.: Symplectic manifolds with no Kähler structures. Lectures Notes in Mathematics, 1661. Springer, Berlin (1997)Google Scholar
  24. 24.
    Turkowski P.: Solvable Lie algebras of dimension six. J. Math. Phys. 31, 1344–1350 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Yamada T.: A pseudo-Kähler structure on a nontoral compact complex parallelizable solvmanifold. Geom. Dedicata 112, 115–122 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Facultad de Ciencia y TecnologíaUniversidad del PaísVascoBilbaoSpain
  2. 2.Departamento de Matemáticas-I.U.M.A.Universidad de ZaragozaZaragozaSpain

Personalised recommendations