Mirror Duality via G2 and Spin(7) Manifolds

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 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

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

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