Mirror Duality via G2 and Spin(7) Manifolds

  • Selman Akbulut
  • Sema Salur
Part of the Progress in Mathematics book series (PM, volume 279)


The main purpose of this chapter is to give a construction of certain “mirror dual” Calabi–Yau submanifolds inside of a G 2 manifold. More specifically, we explain how to assign to a G 2 manifold (M, φ, Λ), with the calibration 3-form φ and an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2- and 3-forms of M. These forms can then be used to define different complex and symplectic structures on certain 6-dimensional subbundles of T(M). When these bundles are integrated they give mirror CY manifolds. In a similar way, one can define mirror dual G 2 manifolds inside of a Spin(7) manifold (N 8, Ψ). In case N 8 admits an oriented 3-plane field, by iterating this process we obtain Calabi–Yau submanifold pairs in N whose complex and symplectic structures determine each other via the calibration form of the ambient G 2 (or Spin(7)) manifold.


Tangent Bundle Symplectic Structure Holonomy Group Bundle Versus Star Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ac]
    B. Acharya, On Mirror Symmetry for Manifolds of Exceptional Holonomy, Nucl.Phys. B524 (1998) 269–282, hep-th/9707186.CrossRefMathSciNetGoogle Scholar
  2. [AV]
    M. Aganagic and C. Vafa, G 2 Manifolds, Mirror Symmetry and Geometric Engineering, hep-th/0110171.Google Scholar
  3. [AS]
    S. Akbulut and S. Salur, Deformations in G 2 manifolds, Adv. Math. 217 (2008), no. 5, 2130–2140.MATHCrossRefMathSciNetGoogle Scholar
  4. [AES]
    S. Akbulut, B. Efe and S. Salur, Mirror Duality in a Joyce Manifold, Adv. Math, 9 September 2009, math.GT/0707.1512.Google Scholar
  5. [ASa]
    V. Apostolov and S. Salamon, Kahler Reduction of metrics with holonomy G 2, math.DG/0303197.Google Scholar
  6. [AW]
    M. Atiyah and E. Witten, M-theory dynamics on a manifold of G 2 holonomy, Adv. Theor. Math. Phys. 6 (2003), 1–106.MathSciNetGoogle Scholar
  7. [B1]
    R.L. Bryant, Metrics with exceptional holonomy, Ann. of Math 126 (1987), 525–576.CrossRefMathSciNetGoogle Scholar
  8. [B2]
    R.L. Bryant, Some remarks on G 2 -structures, Proceedings of Gökova Geometry-Topology Conference 2005, 75–109, Gökova Geometry/Topology Conference (GGT), Gökova, 2006.Google Scholar
  9. [B3]
    R.L. Bryant, Submanifolds and special structures on the Octonions, J. Differential Geom. 17 (1982), no. 2, 185–232.MATHMathSciNetGoogle Scholar
  10. [BS]
    R.L. Bryant and M.S. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. Jour., vol. 58, no 3 (1989), 829–850.MATHCrossRefMathSciNetGoogle Scholar
  11. [C1]
    F.M. Cabrera, SU(3)-structures on hypersurfaces of manifolds withG 2 -structures, Monatsh. Math. 148 (2006), no. 1, 29–50.MATHCrossRefMathSciNetGoogle Scholar
  12. [C2]
    F.M. Cabrera, Orientable hypersurfaces of Riemannian manifolds with Spin(7)-structure, Acta Math. Hungar. 76 (1997), no. 3, 235–247.MATHCrossRefMathSciNetGoogle Scholar
  13. [Ca]
    E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958), 407–438.MATHMathSciNetGoogle Scholar
  14. [CS]
    S. Chiossi and S. Salamon, The intrinsic torsion of SU(3) and G 2 -structures, math.DG/0202282.Google Scholar
  15. [G]
    A. Gray, Vector cross product on manifolds, Trans. Amer. Math. Soc. 141 (1969), 463–504, Correction 148 (1970), 625.Google Scholar
  16. [GM]
    S. Gurrieri and A. Micu Type IIB Theory on half-flat manifolds, Classical Quantum Gravity 20 (2003), no. 11, 2181–2192.Google Scholar
  17. [GYZ]
    S. Gukov, S.T. Yau and E. Zaslow, Duality and Fibrations on G 2 Manifolds, Turkish J. Math. 27 (2003), no. 1, 61–97.MATHMathSciNetGoogle Scholar
  18. [Hi1]
    N.J. Hitchin, The moduli space of special Lagrangian submanifolds , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, 503–515 (1998).Google Scholar
  19. [Hi2]
    N.J. Hitchin, Stable forms and special metrics, Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70–89, Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001.Google Scholar
  20. [HL]
    F.R. Harvey and H.B. Lawson, Calibrated geometries, Acta. Math. 148 (1982), 47–157.MATHCrossRefMathSciNetGoogle Scholar
  21. [IC]
    S. Ivanov and M. Cabrera SU(3)-structures on submanifolds of a Spin(7)-manifold, Differential Geom. Appl. 26 (2008), no. 2, 113–132.Google Scholar
  22. [J]
    D.D. Joyce, Compact Manifolds with special holonomy, OUP, Oxford, 2000.MATHGoogle Scholar
  23. [K]
    S. Karigiannis, Deformations of G 2 and Spin(7) structures on manifolds, Canad. J. Math. 57 (2005), no. 5, 1012–1055.MATHMathSciNetGoogle Scholar
  24. [M]
    R.C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705–747.MATHMathSciNetGoogle Scholar
  25. [Ma]
    M. Marino, Enumerative geometry and knot invariants, Infinite dimensional groups and manifolds, 27–92, IRMA Lect. Math. Theor. Phys., 5, de Gruyter, Berlin, 2004. hep-th/0210145.Google Scholar
  26. [S]
    S. Salamon, Riemannian geometry and holonomy groups, Pitman Res. Notes in Math. Series, no. 201.Google Scholar
  27. [SV]
    S.L. Shatashvili and C. Vafa Superstrings and Manifolds of Exceptional Holonomy, Selecta Math. 1 (1995) 347–381.MATHCrossRefMathSciNetGoogle Scholar
  28. [T]
    E. Thomas, Postnikov invariants and higher order cohomology operations, Ann. Math. (2) 85, (1967) 184–217.Google Scholar
  29. [Th]
    W. Thurston, Existence of codimension-one foliations, Ann. Math. (2) 104, (1976) 249–268.Google Scholar
  30. [Th1]
    W. Thurston, Private communication.Google Scholar
  31. [Ti]
    G. Tian, Gauge Theory and Calibrated Geometry I, Ann. Math. 151, (2000), 193–268.MATHCrossRefGoogle Scholar
  32. [V]
    C. Vafa, Simmons Lectures (Harvard).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Dept. of MathematicsUniversity of RochesterRochesterUSA

Personalised recommendations