Summary
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.
partially supported by NSF grant 0805858 DMS 0505638
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Akbulut, S., Salur, S. (2010). Mirror Duality via G 2 and Spin(7) Manifolds. In: Ceyhan, Ö., Manin, Y.I., Marcolli, M. (eds) Arithmetic and Geometry Around Quantization. Progress in Mathematics, vol 279. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4831-2_1
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