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MIRROR DUALITY VIA G2 AND Spin(7) MANIFOLDS SELMAN AKBULUT AND SEMA SALUR
 

Summary: MIRROR DUALITY VIA G2 AND Spin(7) MANIFOLDS
SELMAN AKBULUT AND SEMA SALUR
Abstract. The main purpose of this paper is to give a construction of certain
"mirror dual" Calabi-Yau submanifolds inside of a G2 manifold. More specifically,
we explain how to assign a G2 manifold (M, , ), with the calibration 3-form
and an oriented 2-plane field , 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 G2 manifolds inside of a Spin(7) manifold (N8
, ). In case N8
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 G2 (or Spin(7)) manifold.
1. Introduction
Let (M7, ) be a G2 manifold with the calibration 3-form . If restricts to be
the volume form of an oriented 3-dimensional submanifold Y 3, then Y is called an
associative submanifold of M. Associative submanifolds are very interesting objects
as they behave very similarly to holomorphic curves of Calabi-Yau manifolds.
In [AS], we studied the deformations of associative submanifolds of (M, ) in order

  

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

 

Collections: Mathematics